Lost-sales dynamics, admissible policies, and the two-variable cost certificate

DemandLaw. A demand-law object consists of a measure $P$ on the nonnegative finite real numbers $[0,\infty)$, a proof that $P$ has total mass one, a proof that the real-valued identity function $d\mapsto d$ is Bochner integrable under $P$, and the strict inequality $0<\int d\,P(\mathrm d d)$. Thus its real mean is finite and strictly positive. There is no assumption of a density, non-atomicity, bounded support, or any higher moment; individual demands may be zero, and a deterministic strictly positive demand law is included. The probability proof is registered for subsequent use as a probability-measure assumption.

mean. For any demand-law object $P$, whose underlying measure is a probability measure on $[0,\infty)$ with integrable identity and strictly positive identity integral, $\operatorname{mean}(P)$ is the real Bochner integral $\mu=\int_{[0,\infty)}d\,P(\mathrm d d)$. The hypotheses carried by $P$ make $\mu$ finite and strictly positive; this definition does not use an extended-real mean.

Parameters. A parameter object $c$ contains a natural number $L$, a proof of $1\le L$, two real numbers $h,p$, and proofs of $0<h$ and $0<p$. In particular the admissible parameter objects have a positive integer lead-time value and strictly positive finite real coefficients; neither $L=0$ nor a zero or negative coefficient is included. No demand law or other relation among these parameters is part of the object.

DemandPath. A demand path is an arbitrary function $d:\mathbb N\to[0,\infty)$, with natural indices starting at zero and each value a finite nonnegative real number. This type includes every such infinite sequence; no probabilistic, convergence, boundedness, or support condition is imposed on an individual path.

Sample. A sample is a pair $\omega=(d,u)$, where $d:\mathbb N\to[0,\infty)$ is an arbitrary demand sequence and $u\in\mathbb R$ is an arbitrary real number. The sample space itself allows every real seed, including values outside $[0,1]$; restriction of the seed to that interval is a property of the sampling measure, not of this type.

demandPathLaw. For a probability measure $P$ on $[0,\infty)$ supplied with an integrable identity and strictly positive mean, the demand-path law is the countable product measure $\bigotimes_{i\in\mathbb N}P$ on $[0,\infty)^{\mathbb N}$, with the usual product measurable structure. Accordingly its coordinate demands all have law $P$ and are independent. The finite-positive-mean assumptions are required by the argument type even though the displayed product construction uses only the underlying measures and their probability structure.

seedLaw. The seed law on $\mathbb R$ is Lebesgue measure restricted to the closed interval $[0,1]$: for a measurable set $A$, its measure is the Lebesgue measure of $A\cap[0,1]$. This is uniform probability on that interval, with both endpoints having mass zero; it is defined on all real seeds.

seedLaw_probability. Lebesgue measure on $\mathbb R$ restricted to $[0,1]$ has total mass one, so it is a probability measure. This assertion has no parameters or additional hypotheses and is registered as a probability-measure instance.

sampleLaw. For a demand-law object $P$ with finite strictly positive real mean, the sampling measure on $([0,\infty)^{\mathbb N})\times\mathbb R$ is $\mathbb Q_P=(\bigotimes_{i\in\mathbb N}P)\otimes(\lambda|_{[0,1]})$, where $\lambda$ is real Lebesgue measure. Thus the demands are independent with common law $P$, and the real seed is uniform on $[0,1]$ and independent of the entire demand sequence. There is one real seed coordinate, not a separately specified sequence of seed coordinates.

demandPathLaw_probability. For every probability demand measure $P$ on finite nonnegative reals that comes with an integrable identity and strictly positive mean, its countably infinite product $\bigotimes_{i\in\mathbb N}P$ is a probability measure on demand sequences. This conclusion is registered as an instance; there are no additional assumptions on $P$.

sampleLaw_probability. For every probability demand measure $P$ on finite nonnegative reals with integrable identity and strictly positive mean, the product of its countable demand-sequence law and Lebesgue measure restricted to $[0,1]$ is a probability measure on pairs consisting of a demand sequence and a real seed. This is an instance assertion about the full sampling measure $\mathbb Q_P$, with no additional assumptions.

history. For every $t\in\mathbb N$ and every sample $\omega=(d,u)\in([0,\infty)^{\mathbb N})\times\mathbb R$, the history at $t$ is the pair $((d_0,\ldots,d_{t-1}),u)$, formally a function on $\{0,\ldots,t-1\}$ together with the original seed. It contains all demands strictly before $t$ and the seed, and contains no demand at or after $t$. When $t=0$, the demand-history domain is empty and only the seed carries variable information.

AdmissiblePolicy. An admissible policy is a family, indexed by every $t\in\mathbb N$, of measurable functions $\pi_t:([0,\infty)^{\{0,\ldots,t-1\}})\times\mathbb R\to[0,\infty)$, using the usual product measurable structures. Each output is a finite nonnegative real order. The input is the entire past demand vector and one real seed; no condition of stationarity, bounded order size, integrable orders or costs, or any prescribed dependence on a state is imposed. Measurability is required separately for each time, and the functions are defined on every possible history and every real seed. At $t=0$ a decision can depend on the seed and has no past demands to use.

order. For every admissible policy $\pi$, time $t\in\mathbb N$, and sample $\omega=(d,u)$, the order is $q_t(\omega)=\pi_t((d_0,\ldots,d_{t-1}),u)\in[0,\infty)$. Here each $\pi_t$ is a measurable function of exactly that finite past-demand vector and the real seed. This evaluation uses no present or future demand; in particular $q_0$ is evaluated on the empty demand vector and $u$. No demand-law or parameter argument is required.

State. For every natural number $L$, a state is a pair $(x,a)$ with $x\in[0,\infty)$ and $a:\{0,\ldots,L-1\}\to[0,\infty)$; all entries are finite. This type itself allows $L=0$, in which case the vector has empty domain, as well as every nonnegative value of $x$ and every nonnegative vector. It imposes no bound or reachability condition.

initialState. For every $L\in\mathbb N$, the initial state is $(0,(0)_{i=0}^{L-1})$: its scalar coordinate and every vector coordinate are zero. This is defined also at $L=0$, where the vector is the unique empty function.

arrival. Given parameters $c=(L,h,p)$ with $L\in\mathbb N$, $L\ge1$, and $h,p>0$, and any state $(x,a)\in[0,\infty)\times[0,\infty)^{\{0,\ldots,L-1\}}$, the arrival is $a_0$. The positive-lead-time proof supplies the existence of coordinate zero. The value is finite and nonnegative, and $h,p,x$ do not enter its formula.

step. Given parameters $c=(L,h,p)$ with $L\ge1$ and $h,p>0$, any state $(x,a)$ with nonnegative finite coordinates, and any finite nonnegative order $q$ and demand $d$, one step returns $(x',a')$, where $x'=(x+a_0-d)^+$ and $a'_i=a_{i+1}$ if $i+1<L$, while $a'_i=q$ otherwise, for every $i\in\{0,\ldots,L-1\}$. Here $v^+=\max(v,0)$: subtraction in the nonnegative-real type is truncated. The final vector coordinate is therefore $q$, and all earlier coordinates shift down by one. For $L=1$ the new vector consists only of $q$. The current order does not enter $x'$; only the old vector's first coordinate does. The positive real coefficients $h,p$ are carried by the parameter object but do not affect this transition.

run. For parameters $c=(L,h,p)$ with $L\ge1$ and $h,p>0$, any admissible family of measurable decisions $\pi_t$, and every sample $\omega=(d,u)$, the state sequence $s_t=(x_t,a_t)$ is defined for all $t\in\mathbb N$ by $x_0=0$, $a_{0,i}=0$, and $x_{t+1}=(x_t+a_{t,0}-d_t)^+$, $a_{t+1,i}=a_{t,i+1}$ for $i+1<L$, and $a_{t+1,L-1}=\pi_t((d_0,\ldots,d_{t-1}),u)$. Every demand and seed here is arbitrary at the pointwise level; no sampling measure is needed to define the recursion. In particular the period-$t$ order is inserted in the state at $t+1$ and enters the available amount used with demand $d_{t+L}$; $d_t$ is processed using $s_t$ and its existing first vector coordinate. All shortages are truncated to zero in the scalar coordinate.

filledDemand. For parameters $c=(L,h,p)$ with positive integer $L$ and strictly positive real $h,p$, any nonnegative state $(x,a)$ of length $L$, and any finite nonnegative demand $d$, the filled-demand value is $\min(d,x+a_0)\in[0,\infty)$. It uses the scalar state plus the existing first vector coordinate and is defined for arbitrary states, not only states reached by a policy; no current order is an argument.

lostSales. For parameters $c=(L,h,p)$ with $L\ge1$ and $h,p>0$, any state $(x,a)$ with finite nonnegative coordinates, and any finite nonnegative demand $d$, lost sales are $(d-(x+a_0))^+\in[0,\infty)$. The subtraction is truncated at zero, so this value is zero whenever demand is at most $x+a_0$. The definition does not carry a negative inventory balance forward.

periodCost. For parameters $c=(L,h,p)$ with positive integer $L$ and strictly positive finite real $h,p$, a state $(x,a)$ with finite nonnegative coordinates, and finite nonnegative demand $d$, the period cost is the extended nonnegative real $h(x+a_0-d)^++p(d-x-a_0)^+$. More literally, each coefficient is embedded by $v\mapsto\max(v,0)$ into $[0,\infty]$, and each truncated nonnegative-real quantity is embedded there before multiplication and addition; positivity of $h,p$ means those coefficient embeddings do not change their values. Each such pointwise cost is finite. The two terms use the amount remaining after demand and the demand left unfilled, respectively; no cost of the order itself or separate vector-coordinate cost occurs in this formula.

averageCost. For a probability demand law $P$ on finite nonnegative reals with integrable identity and strictly positive mean, and an arbitrary function $C:\mathbb N\times(([0,\infty)^{\mathbb N})\times\mathbb R)\to[0,\infty]$, define $\operatorname{averageCost}_P(C)=\limsup_{T\to\infty}\frac{1}{T+1}\sum_{t=0}^{T}\int^- C(t,\omega)\,\mathbb Q_P(\mathrm d\omega)$, where $\mathbb Q_P=(\bigotimes_{i\in\mathbb N}P)\otimes(\lambda|_{[0,1]})$ and the limsup is along natural $T$. Each integral is the extended nonnegative lower Lebesgue integral; the argument $C$ is not required to be measurable or integrable, so the definition uses the total lower-integral construction even for a nonmeasurable section. Infinite integrals and the value $+\infty$ are allowed. There are $T+1$ periods starting at zero, including just period zero when $T=0$; the divisor is always a finite strictly positive extended nonnegative real. The limsup is taken after summing the separate timewise integrals and normalizing, not on samplewise average costs.

policyCost. For a probability demand law $P$ on $[0,\infty)$ with finite strictly positive mean, parameters $c=(L,h,p)$ with $L\ge1$ and $h,p>0$, and an admissible policy consisting of measurable maps $\pi_t$ from past demand vectors and a real seed to finite nonnegative orders, its cost is $\limsup_{T\to\infty}(T+1)^{-1}\sum_{t=0}^{T}\int^-[h(x_t+a_{t,0}-d_t)^++p(d_t-x_t-a_{t,0})^+]\,\mathbb Q_P(\mathrm d(d,u))$. Here $\mathbb Q_P$ is the independent product of i.i.d. demand coordinates with law $P$ and a uniform $[0,1]$ seed; $x_0=0$, $a_{0,i}=0$, $x_{t+1}=(x_t+a_{t,0}-d_t)^+$, earlier vector coordinates shift left, and $a_{t+1,L-1}=\pi_t((d_0,\ldots,d_{t-1}),u)$. The sum and lower integrals take values in $[0,\infty]$, so pointwise finite period costs need not give a finite value for this definition; no finite-cost assumption is placed on the policy.

OPT. For every probability demand law $P$ on $[0,\infty)$ with finite strictly positive mean and every parameter tuple $(L,h,p)$ with integer $L\ge1$ and $h,p>0$, $\mathrm{OPT}(P,c)$ is the infimum in $[0,\infty]$ over all families of measurable maps $\pi_t:[0,\infty)^{\{0,\ldots,t-1\}}\times\mathbb R\to[0,\infty)$ of their limsup expected average cost. For each such family, the state starts at zero, period-$t$ order is $\pi_t(d_{<t},u)$, the next scalar state is $(x_t+a_{t,0}-d_t)^+$, the vector shifts left and appends the order, and the period cost is $h(x_t+a_{t,0}-d_t)^++p(d_t-x_t-a_{t,0})^+$. The average is $\limsup_{T\to\infty}(T+1)^{-1}\sum_{t=0}^{T}\int^- (\text{period cost})\,\mathrm d\mathbb Q_P$, with i.i.d. law-$P$ demands and an independent uniform $[0,1]$ seed. Orders are finite and nonnegative but not uniformly bounded or required to be integrable. The set of policies includes the identically zero policy; the definition is an infimum, with no claim that a minimizing policy exists.

cbsOrder. For every natural $L$, finite nonnegative numbers $S,r$, and any state $(x,a)\in[0,\infty)\times[0,\infty)^{\{0,\ldots,L-1\}}$, the order is $Q_{L,S,r}(x,a)=\min\bigl((S-x-\sum_{i=0}^{L-1}a_i)^+,r\bigr)$. The sum includes all vector coordinates, including coordinate zero when it exists, and the subtraction is truncated before taking the minimum. There is no condition $r\le S$ or relation to a demand mean. Either $S=0$ or $r=0$ makes this order zero for every state. The definition also allows $L=0$, when the vector sum is zero and the formula is $\min((S-x)^+,r)$.

cbsRun. For parameters $(L,h,p)$ with integer $L\ge1$ and $h,p>0$, any finite nonnegative $S,r$, and any sample $(d,u)$, the sequence $s_t=(x_t,a_t)$ starts with all coordinates zero and satisfies $q_t=\min((S-x_t-\sum_{i=0}^{L-1}a_{t,i})^+,r)$, $x_{t+1}=(x_t+a_{t,0}-d_t)^+$, $a_{t+1,i}=a_{t,i+1}$ for $i+1<L$, and $a_{t+1,L-1}=q_t$, for all $t\in\mathbb N$. Thus the order is computed from the state at $t$, whose vector still contains its first coordinate, and is used as an arrival with demand $d_{t+L}$. The recursion ignores the seed $u$ and places no probabilistic condition on the demand path. In particular its first order is $\min(S,r)$; if $S=0$ or $r=0$, all orders and state coordinates remain zero.

cbsCost. For a probability demand law $P$ on finite nonnegative reals with finite strictly positive mean, parameters $(L,h,p)$ with $L\ge1$ and $h,p>0$, and any finite nonnegative $S,r$, the cost is $\limsup_{T\to\infty}(T+1)^{-1}\sum_{t=0}^{T}\int^-[h(x_t+a_{t,0}-d_t)^++p(d_t-x_t-a_{t,0})^+]\,\mathbb Q_P(\mathrm d(d,u))$. Here demands are i.i.d. with law $P$, the seed is independent uniform on $[0,1]$, all state coordinates start at zero, $q_t=\min((S-x_t-\sum_i a_{t,i})^+,r)$, the scalar state updates to $(x_t+a_{t,0}-d_t)^+$, and the vector shifts left and appends $q_t$. The integral is the extended nonnegative lower integral, the resulting cost is defined in $[0,\infty]$, and the state and integrand ignore $u$. Neither a strict positive cap nor a strict positive target nor a relation between $r$ and the mean is assumed.

CbsOpt. For a probability demand law $P$ on $[0,\infty)$ with finite strictly positive mean and parameters $(L,h,p)$ with positive integer $L$ and $h,p>0$, this quantity is the nested extended-nonnegative-real infimum $\inf_{S\in[0,\infty)}\inf_{r\in[0,\infty)}C_{P,c}(S,r)$. For each pair, $C_{P,c}(S,r)$ is the limsup of the mean of periodwise lower integrals over periods $0,\ldots,T$, under i.i.d. law-$P$ demands and an independent uniform $[0,1]$ seed, of $h(x_t+a_{t,0}-d_t)^++p(d_t-x_t-a_{t,0})^+$; states start at zero, update their scalar coordinate by $(x_t+a_{t,0}-d_t)^+$, and shift the length-$L$ vector left while appending $\min((S-x_t-\sum_i a_{t,i})^+,r)$. Both optimization variables range over all finite nonnegative reals, including zero, without a constraint involving one another or the demand mean. This is an infimum and does not assert attainment.

baseStockCost. For a probability demand law $P$ on $[0,\infty)$ with finite strictly positive mean, parameters $(L,h,p)$ with $L\ge1$ and $h,p>0$, and any finite nonnegative $S$, this is the cost obtained by setting both the target and the cap equal to $S$ in the preceding capped-order recursion. Explicitly, all state coordinates start at zero, $q_t=\min((S-x_t-\sum_i a_{t,i})^+,S)$, $x_{t+1}=(x_t+a_{t,0}-d_t)^+$, and the length-$L$ vector shifts left and appends $q_t$. The cost is $\limsup_{T\to\infty}(T+1)^{-1}\sum_{t=0}^{T}\int^-[h(x_t+a_{t,0}-d_t)^++p(d_t-x_t-a_{t,0})^+]\,\mathrm d\mathbb Q_P$, where demands are i.i.d. with law $P$ and the seed is independent uniform on $[0,1]$. Because all state coordinates are nonnegative, the minimum with $S$ cannot reduce the displayed positive-part shortfall; $S=0$ is included and gives zero orders.

finiteInventory. For every arbitrary demand sequence $d:\mathbb N\to[0,\infty)$, every real $r$ (including negative values), and every natural $m$ (including zero), this real-valued function is $I_m(d,r)=\max_{0\le k\le m}\sum_{i=0}^{k-1}(r-d_i)$. The maximum is over the nonempty finite set of integers $0,\ldots,m$, and the $k=0$ sum is zero. In particular $I_m(d,r)\ge0$ and $I_0(d,r)=0$. The sum consists of prefixes starting at index zero; it involves only $d_0,\ldots,d_{m-1}$ and does not maximize over a starting index or over an infinite horizon. No measure, mean, or stability assumption occurs in this definition.

certificateExcess. For every demand sequence $d:\mathbb N\to[0,\infty)$, arbitrary real $r,z$, and natural $m$, this is the extended nonnegative real $E_m(d;r,z)=\left(\max_{0\le k\le m}\sum_{i=0}^{k-1}(r-d_i)+\sum_{i=0}^{m-1}(d_i-r)-z\right)^+$ embedded in $[0,\infty]$. The maximum includes the zero prefix, both sums are finite, and no nonnegativity or mean restriction is imposed on $r$ or $z$ here. Thus every pointwise value is finite, even though later lower integrals use an extended codomain. At $m=0$ this is $(-z)^+$, independent of the demand path and of $r$.

CertificateFeasible. Let $P$ be a probability law on finite nonnegative reals with finite strictly positive real mean $\mu=\int d\,P(\mathrm d d)$, let $(L,h,p)$ satisfy $L\in\mathbb N$, $L\ge1$, and $h,p>0$, and let $r,z$ be arbitrary real numbers. The predicate holds exactly when all five conditions hold: $0\le r$, $r\le\mu$, $0\le z$, $\int^- E_L(d;r,z)\,\mathrm d(P^{\otimes\mathbb N})\le (L(\mu-r))^+$, and $\int^- E_{L+1}(d;r,z)\,\mathrm d(P^{\otimes\mathbb N})\le ((L+1)(\mu-r))^+$. Here $E_m(d;r,z)=\left(\max_{0\le k\le m}\sum_{i=0}^{k-1}(r-d_i)+\sum_{i=0}^{m-1}(d_i-r)-z\right)^+$, each right-hand side is embedded in $[0,\infty]$, and the integrals are extended nonnegative lower integrals over the canonical i.i.d. demand-path measure, without a seed. The first two conditions make these right-hand sides finite and nonnegative without truncation, so feasibility requires finite integrals at both specified horizons. The tests are only at $L$ and $L+1$, not at all horizons. Both endpoints $r=0$ and $r=\mu$ and the endpoint $z=0$ are permitted; at $r=\mu$ both integral bounds are zero. The coefficients $h,p$ satisfy the carried parameter assumptions but do not enter the feasibility inequalities.

certificateObjective. For a probability demand law $P$ on finite nonnegative reals with finite strictly positive real mean $\mu$, parameters $(L,h,p)$ with integer $L\ge1$ and $h,p>0$, and arbitrary real $r,z$, the objective is the extended nonnegative real $(hz+p(\mu-r))^+$. The positive part is applied to the whole real sum before embedding it in $[0,\infty]$; it is not applied to the two summands separately. This definition does not assume feasibility, $r\ge0$, $r\le\mu$, or $z\ge0$, so it also assigns zero to any inputs making the sum nonpositive. Each value is finite. For feasible inputs the two summands are nonnegative and the positive part leaves their sum unchanged.

lowerCertificate. For a probability demand law $P$ on finite nonnegative reals with finite strictly positive mean $\mu$, and parameters $(L,h,p)$ with integer $L\ge1$ and $h,p>0$, this is the infimum in $[0,\infty]$ of $(hz+p(\mu-r))^+$ over real pairs $(r,z)$ such that $0\le r\le\mu$, $z\ge0$, and, for each of the two values $m=L,L+1$, $\int^-\left(\max_{0\le k\le m}\sum_{i=0}^{k-1}(r-d_i)+\sum_{i=0}^{m-1}(d_i-r)-z\right)^+\,\mathrm d(P^{\otimes\mathbb N})\le m(\mu-r)$. The right-hand side is embedded in the extended nonnegative reals, and the integral is the lower integral over i.i.d. demand sequences. The indexing is by the subtype of pairs satisfying all these conditions, with no minimizer or separate existence witness required as an argument. In this codomain an infimum over an empty index type would be $+\infty$, but here $(r,z)=(0,0)$ satisfies both tests: the maximum prefix is zero, the excess is the sum of the first $m$ demands, and its integral is $m\mu$. Its objective is $p\mu$. A strictly positive value is not required: for a law concentrated at the positive number $\mu$, the feasible pair $(\mu,0)$ has zero excess and zero objective, and this infimum is zero. No attainment or comparison to any policy cost is asserted.

rho. For every natural $L$, independently of any parameter object or demand law, $\rho(L)$ is the real number $L/(L+1)$, with the naturals cast to reals before arithmetic. This includes $L=0$, for which $\rho(0)=0$; its denominator is nonzero for every allowed input.

kappa. For every natural $L$, independently of any parameter object or demand law, $\kappa(L)$ is the real number $1+\frac{4L^2}{(L+1)(3L-1)}$, with subtraction, multiplication, exponentiation, and division performed in the reals after casting $L$. The denominator uses the real expression $3L-1$, not truncated natural subtraction. The domain includes $L=0$, where the denominator is $-1$ and $\kappa(0)=1$; for integer $L\ge1$ the denominator is positive, so it is never zero on the stated domain.

completeHistory. For every $t\in\mathbb N$ and every pair $a=(v,u)$ with $v:\{0,\ldots,t-1\}\to[0,\infty)$ and $u\in\mathbb R$, the completed sample is $(\widetilde d,u)$, where $\widetilde d_k=v_k$ if $k<t$ and $\widetilde d_k=0$ if $k\ge t$. Thus the supplied finite demand vector is extended by zeros from index $t$ onward, with the seed preserved even when it lies outside $[0,1]$. At $t=0$ the entire completed demand sequence is zero. The construction has no law or support requirement.

measurable_completeHistory. For each natural $t$, the map from a finite nonnegative demand vector $v:\{0,\ldots,t-1\}\to[0,\infty)$ and real seed $u$ to the sample $(\widetilde d,u)$, where $\widetilde d_k=v_k$ for $k<t$ and $\widetilde d_k=0$ for $k\ge t$, is measurable for the usual product measurable structures. This holds also at $t=0$ and on all real seeds; it has no probability-law, integrability, or parameter hypotheses.

measurable_cbsRun. For every parameter tuple $(L,h,p)$ with positive integer $L$ and $h,p>0$, every finite nonnegative $S,r$, and every natural $t$, the map from an arbitrary sample $(d,u)\in[0,\infty)^{\mathbb N}\times\mathbb R$ to the state $(x_t,a_t)$ is measurable, where all coordinates start at zero and, at each index $j$, $q_j=\min((S-x_j-\sum_{i=0}^{L-1}a_{j,i})^+,r)$, $x_{j+1}=(x_j+a_{j,0}-d_j)^+$, earlier vector coordinates shift left, and $a_{j+1,L-1}=q_j$. The codomain is the scalar nonnegative-real coordinate times the length-$L$ nonnegative vector with its product measurable structure. This assertion includes $t=0$, $S=0$, and $r=0$ and does not require a demand law or a restricted seed value.

measurable_cbsOrder. For each natural $L$ and all finite nonnegative $S,r$, the function on all nonnegative states $(x,a)\in[0,\infty)\times[0,\infty)^{\{0,\ldots,L-1\}}$ given by $(x,a)\mapsto\min((S-x-\sum_{i=0}^{L-1}a_i)^+,r)$ is measurable. This uses the usual product measurable structure and requires no lead-time positivity, demand law, or state reachability hypothesis. At $L=0$ the sum is empty and zero; zero targets and zero caps are included.

cbsPolicy. For each parameter tuple $(L,h,p)$ with positive integer $L$ and $h,p>0$, and each finite nonnegative $S,r$, this constructs an admissible family of measurable decisions as follows. At time $t\in\mathbb N$, given a finite history $v:\{0,\ldots,t-1\}\to[0,\infty)$ and real seed $u$, extend $v$ to a demand path by assigning zero at every index $k\ge t$, retain $u$, and run from zero for $t$ steps with orders $q_j=\min((S-x_j-\sum_{i=0}^{L-1}a_{j,i})^+,r)$, scalar updates $x_{j+1}=(x_j+a_{j,0}-d_j)^+$, and vector updates that shift left and append $q_j$. Output $\min((S-x_t-\sum_i a_{t,i})^+,r)$. The construction supplies measurability of every decision by composition of the measurable history completion, state recursion, and order map. It is defined for every history and real seed, and in fact its output does not depend on that seed; at $t=0$ it outputs $\min(S,r)$.

cbsRun_congr_history. For every parameter tuple $(L,h,p)$ with positive integer $L$ and $h,p>0$, all finite nonnegative $S,r$, any natural $t$, and any two samples $\omega=(d,u)$ and $\eta=(e,v)$, if $d_k=e_k$ for every natural $k<t$, then their states at time $t$ coincide under the following recursion: start at zero, set each order to $\min((S-x_j-\sum_i a_{j,i})^+,r)$, update the scalar state to $(x_j+a_{j,0}-d_j)^+$ or $(x_j+a_{j,0}-e_j)^+$ respectively, and shift the length-$L$ vector left while appending the order. No equality of seeds or of demands at or after $t$ is required. At $t=0$ the hypothesis is vacuous and both states are the zero state. This is a pointwise equality for all samples, not an almost-everywhere assertion under any measure.

cbsRun_completeHistory. For every parameter tuple $(L,h,p)$ with positive integer $L$ and $h,p>0$, all finite nonnegative $S,r$, every natural $t$, and every sample $(d,u)$, replacing its demand path by $\widetilde d_k=d_k$ for $k<t$ and $\widetilde d_k=0$ for $k\ge t$, while preserving $u$, does not change the state at time $t$ of the recursion that starts at zero, orders $\min((S-x_j-\sum_i a_{j,i})^+,r)$, updates the scalar state to $(x_j+a_{j,0}-d_j)^+$, and shifts the length-$L$ vector left and appends that order. The same recursion is used with $\widetilde d$ on the other side. This holds pointwise for all paths and real seeds, including $t=0$, and requires no sampling law or almost-everywhere qualification.

run_cbsPolicy. For every parameter tuple $(L,h,p)$ with positive integer $L$ and $h,p>0$, all finite nonnegative $S,r$, every sample $(d,u)$, and every natural $t$, two zero-initialized state constructions have exactly the same state at time $t$. The first uses the admissible decision which, at each time $j$, completes the observed vector $(d_0,\ldots,d_{j-1})$ by zeros, runs the state-based recursion for $j$ steps on that completed sample, and returns $\min((S-x_j-\sum_i a_{j,i})^+,r)$ for that reconstructed state. The second directly returns this same order formula from its current state on the original sample. In both constructions the scalar update is $(x_j+a_{j,0}-d_j)^+$ and the length-$L$ vector shifts left and appends the chosen order. Equality is pointwise for every time and every real seed, including $t=0$, without a demand-law hypothesis.

policyCost_cbsPolicy. For every probability demand law $P$ on finite nonnegative reals with finite strictly positive mean, every parameter tuple $(L,h,p)$ with positive integer $L$ and $h,p>0$, and all finite nonnegative $S,r$, the extended average cost of the admissible history-based policy constructed by zero-completing the past and reconstructing the capped-order state equals the extended average cost of the direct capped-order state recursion. More explicitly, the reconstructed policy at time $j$ evaluates $\min((S-x_j-\sum_i a_{j,i})^+,r)$ after running from zero on the zero-completed observed demands; the direct recursion evaluates that formula on its current state. Both use scalar update $(x_j+a_{j,0}-d_j)^+$ and a length-$L$ vector that shifts left and appends the order. Each cost is $\limsup_{T\to\infty}(T+1)^{-1}\sum_{t=0}^{T}\int^-[h(x_t+a_{t,0}-d_t)^++p(d_t-x_t-a_{t,0})^+]\,\mathrm d\mathbb Q_P$, with i.i.d. law-$P$ demands and an independent uniform $[0,1]$ seed. The assertion is equality in $[0,\infty]$, with no extra finiteness or positivity requirement on $S,r$.

OPT_le_CbsOpt. For every probability demand law $P$ on finite nonnegative reals with finite strictly positive mean and every parameter tuple $(L,h,p)$ with positive integer $L$ and $h,p>0$, the infimum over all measurable history-and-seed policies is less than or equal to the infimum over all finite nonnegative targets $S$ and caps $r$ of the costs of the state-based order formula $\min((S-x_t-\sum_i a_{t,i})^+,r)$. The larger policy class consists of all measurable maps $\pi_t:[0,\infty)^{\{0,\ldots,t-1\}}\times\mathbb R\to[0,\infty)$ for each $t\in\mathbb N$. Every cost uses zero initial state, the scalar update $(x_t+a_{t,0}-d_t)^+$, and the length-$L$ vector update that shifts left and appends the chosen order; it is the limsup over $T\in\mathbb N$ of $(T+1)^{-1}\sum_{t=0}^{T}\int^-[h(x_t+a_{t,0}-d_t)^++p(d_t-x_t-a_{t,0})^+]\,\mathrm d\mathbb Q_P$, where demands are i.i.d. with law $P$ and the seed is independent uniform on $[0,1]$. The comparison is $\mathrm{OPT}(P,c)\le\mathrm{CbsOpt}(P,c)$ in $[0,\infty]$, includes zero target or cap, and requires no optimizer to exist.