Definitions for On Cohomology theory for topological groups

QAlgorithms.Papers.Independent.Intakethird100.P08.ContinuousModule

For a topological group $G$ and a topological abelian group $A$, a continuous module structure consists of an additive automorphism $\alpha_g:A\to A$ for each $g\in G$, such that $\alpha_1=\operatorname{id}_A$, $\alpha_{gh}=\alpha_g\circ\alpha_h$ for every $g,h\in G$, and the map $G\times A\to A$ given by $(g,a)\mapsto\alpha_g(a)$ is continuous.

QAlgorithms.Papers.Independent.Intakethird100.P08.ExtensionPoint

Let $G$ be a Hausdorff, locally compact, second-countable topological group and $A$ a Hausdorff, locally compact, second-countable topological abelian group, both with their Borel $\sigma$-algebras. Fix data $D$ consisting of additive automorphisms $\alpha_g$ of $A$ with $\alpha_1=\operatorname{id}_A$, $\alpha_{gh}=\alpha_g\circ\alpha_h$, and continuous map $(g,a)\mapsto\alpha_g(a)$; a measurable function $\beta:G\times G\to A$ satisfying $\alpha_g(\beta(h,k))+\beta(g,hk)=\beta(g,h)+\beta(gh,k)$ for all $g,h,k\in G$, and $\beta(1,g)=\beta(g,1)=0$ for all $g\in G$; and a specified open neighborhood $N$ of $1$ on whose square $\beta$ is continuous. An extension point for $D$ consists of exactly two coordinates: an arbitrary coefficient $a\in A$ and an arbitrary base element $g\in G$.

QAlgorithms.Papers.Independent.Intakethird100.P08.IsCrossedHom

Let $G$ be a topological group, let $A$ be a topological abelian group, and let $\rho$ specify additive automorphisms $\alpha_g$ of $A$ satisfying $\alpha_1=\operatorname{id}_A$ and $\alpha_{gh}=\alpha_g\circ\alpha_h$, with $(g,a)\mapsto\alpha_g(a)$ continuous. For an arbitrary function $c:G\to A$, the predicate says that, for every $s,t\in G$,

$$c(st)=c(s)+\alpha_s(c(t)).$$

It imposes no continuity or measurability condition on $c$.

QAlgorithms.Papers.Independent.Intakethird100.P08.IsLeftInvariantMeasure

Let $G$ be a Hausdorff, locally compact, second countable topological group, and let $A$ be a Hausdorff, locally compact, second countable additive abelian topological group, both with their Borel measurable structures. Let $D$ consist of additive automorphisms $\alpha_p$ with $\alpha_1=\mathrm{id}$, $\alpha_{pq}=\alpha_p\circ\alpha_q$, and jointly continuous evaluation; a measurable function $\kappa:G\times G\to A$ satisfying $\alpha_p(\kappa(q,r))+\kappa(p,qr)=\kappa(p,q)+\kappa(pq,r)$ and $\kappa(1,p)=\kappa(p,1)=0$; and an open neighborhood $U_0$ of $1$ on whose square $\kappa$ is continuous. Put $E=A\times G$, define $(a,p)\star(b,q)=(a+\alpha_p(b)+\kappa(p,q),pq)$, and equip $E$ with the topology generated by $x\star(U_A\times U_G)$, where $x\in E$, $U_A$ is open and contains $0$, $U_G$ is open and contains $1$, and $\kappa$ is continuous on $U_G^2$. Equip $E$ with the Borel measurable structure of this topology. For a measure $\mu$ on $E$, this predicate means that $\mu$ is not the zero measure; that for every $x\in E$ and every measurable set $W\subseteq E$ whose closure is compact,

$$\mu\bigl(\{x\star w:w\in W\}\bigr)=\mu(W);$$

and that $\mu(K)<+\infty$ for every compact set $K\subseteq E$. The invariance condition is quantified over measurable sets with compact closure, and the compact-set finiteness condition has no separate measurability hypothesis on $K$. Empty sets are included in both quantifications.

QAlgorithms.Papers.Independent.Intakethird100.P08.IsLocalSteinhausSet

Let $G$ be a Hausdorff, locally compact, second countable topological group, and let $A$ be a Hausdorff, locally compact, second countable topological abelian group, both equipped with their Borel measurable structures. Let $D$ specify additive automorphisms $\rho_g$ of $A$ with $\rho_1=\mathrm{id}$, $\rho_{gh}=\rho_g\circ\rho_h$, and jointly continuous $(g,a)\mapsto\rho_g(a)$; a measurable function $c:G\times G\to A$ satisfying $\rho_g(c(h,k))+c(g,hk)=c(g,h)+c(gh,k)$ and $c(1,g)=c(g,1)=0$; and a specified open neighborhood $N$ of $1$ on whose square $c$ is continuous. Put $E=A\times G$, define $(a,g)\star(b,h)=(a+\rho_g(b)+c(g,h),gh)$ and $(a,g)^\dagger=(\rho_{g^{-1}}(-(a+c(g,g^{-1}))),g^{-1})$, and equip $E$ with the topology generated by all sets $\{x\star(a,g):a\in P,\ g\in Q\}$, where $x\in E$, $P$ is an open neighborhood of $0$ in $A$, $Q$ is an open neighborhood of $1$ in $G$, and $c$ is continuous on $Q\times Q$; use the Borel measurable structure of this topology. For any measure $\mu$ on $E$, any $U\subseteq G$, and any $M\subseteq E$, the property defined here holds exactly when all the following conditions hold: $M$ is measurable; for every $x\in E$, $x\in M$ if and only if $x^\dagger\in M$; every $(a,g)\in M$ has $g\in U$; $U$ is open and contains $1$; for every $g\in G$, $g\in U$ if and only if $g^{-1}\in U$; $\overline U$ is compact; $gh\in N$ for every $g,h\in\overline U$; $(0,1)\in M$; and $0<\mu(M)<\infty$. Thus the property excludes empty or measure-zero $M$; it imposes no separate invariance assumption on $\mu$.

QAlgorithms.Papers.Independent.Intakethird100.P08.IsLocallyContinuousMeasurable

For a function $c:G\to A$, with the given topologies and measurable structures on $G$ and $A$ and identity element $1\in G$, the predicate says that $c$ is measurable and that there exists an open set $U\subseteq G$ containing $1$ such that the restriction $c|_U$ is continuous. In particular, the witnessing set $U$ cannot be empty.

QAlgorithms.Papers.Independent.Intakethird100.P08.IsLocallySplitSequence

For identity-preserving multiplicative homomorphisms $i:K\to E$ and $p:E\to Q$, with the given topologies on these spaces, the predicate says that $i$ and $p$ are continuous; $i$ is a closed embedding, meaning that it identifies $K$ homeomorphically with a closed subset of $E$; $p$ sends open sets to open sets and is surjective; and

$$i(K)=\{x\in E:p(x)=1\}.$$

It also requires an open set $U\subseteq Q$ containing $1$ and a continuous map $\sigma:U\to E$, where $U$ has its subspace topology, such that $p(\sigma(u))=u$ for every $u\in U$.

QAlgorithms.Papers.Independent.Intakethird100.P08.IsNonnegativeTest

Let $G$ be a Hausdorff, locally compact, second countable topological group, and let $A$ be a Hausdorff, locally compact, second countable additive abelian topological group, both equipped with their Borel measurable structures. The datum $D$ consists of additive automorphisms $\alpha_p:A\to A$, with $\alpha_1=\mathrm{id}$, $\alpha_{pq}=\alpha_p\circ\alpha_q$, and jointly continuous evaluation $(p,a)\mapsto\alpha_p(a)$; a measurable function $\kappa:G\times G\to A$ satisfying $\alpha_p(\kappa(q,r))+\kappa(p,qr)=\kappa(p,q)+\kappa(pq,r)$ and $\kappa(1,p)=\kappa(p,1)=0$; and an open set $U_0$ containing $1$ such that $\kappa$ is continuous on $U_0\times U_0$. Write $E=A\times G$ and define $(a,p)\star(b,q)=(a+\alpha_p(b)+\kappa(p,q),pq)$. Equip $E$ with the topology generated by the sets $x\star(U_A\times U_G)$, where $x\in E$, $U_A\subseteq A$ is open and contains $0$, $U_G\subseteq G$ is open and contains $1$, and $\kappa$ is continuous on $U_G\times U_G$. For any function $f:E\to\mathbb R$, the predicate in this declaration means that $f$ is continuous, the closure of $\{s\in E:f(s)\ne0\}$ is compact, and $f(s)\ge0$ for every $s\in E$. The zero function is included.

QAlgorithms.Papers.Independent.Intakethird100.P08.IsPositiveInvariantIntegral

Let $G$ be a Hausdorff, locally compact, second countable topological group, and let $A$ be a Hausdorff, locally compact, second countable additive abelian topological group, both with their Borel measurable structures. Let $D$ specify additive automorphisms $\alpha_p$, with $\alpha_1=\mathrm{id}$, $\alpha_{pq}=\alpha_p\circ\alpha_q$, and jointly continuous evaluation; a measurable function $\kappa:G\times G\to A$ satisfying $\alpha_p(\kappa(q,r))+\kappa(p,qr)=\kappa(p,q)+\kappa(pq,r)$ and $\kappa(1,p)=\kappa(p,1)=0$; and an open neighborhood $U_0$ of $1$ such that $\kappa$ is continuous on $U_0^2$. Put $E=A\times G$, define $(a,p)\star(b,q)=(a+\alpha_p(b)+\kappa(p,q),pq)$, and give $E$ the topology generated by $x\star(U_A\times U_G)$ for $x\in E$, open $U_A\ni0$, and open $U_G\ni1$ such that $\kappa$ is continuous on $U_G^2$. Let $C_c(E,\mathbb R)$ denote the real vector space of continuous real functions whose nonzero sets have compact closures. For a real-linear map $I:C_c(E,\mathbb R)\to\mathbb R$, this predicate means that $I$ is not the zero map; that $I(f)\ge0$ whenever $f(s)\ge0$ for every $s\in E$; and that for every $f\in C_c(E,\mathbb R)$ and every $x\in E$, there exists $f_x\in C_c(E,\mathbb R)$ such that $f_x(s)=f(x\star s)$ for every $s\in E$ and $I(f_x)=I(f)$.

QAlgorithms.Papers.Independent.Intakethird100.P08.LocalCocycle

Let $G$ be a Hausdorff, locally compact, second-countable topological group and $A$ a Hausdorff, locally compact, second-countable topological abelian group, each equipped with its Borel $\sigma$-algebra. A local cocycle consists of additive automorphisms $\alpha_g:A\to A$ satisfying $\alpha_1=\operatorname{id}_A$ and $\alpha_{gh}=\alpha_g\circ\alpha_h$, with $(g,a)\mapsto\alpha_g(a)$ continuous; a measurable function $\beta:G\times G\to A$ satisfying, for every $g,h,k\in G$,

$$\alpha_g(\beta(h,k))+\beta(g,hk)=\beta(g,h)+\beta(gh,k);$$

the identities $\beta(1,g)=0$ and $\beta(g,1)=0$ for every $g\in G$; and a specified open set $N\subseteq G$ containing $1$ such that $\beta$ is continuous on $N\times N$.

QAlgorithms.Papers.Independent.Intakethird100.P08.actionTwistedMul

Let $G$ be a Hausdorff, locally compact, second-countable topological group and $A$ a Hausdorff, locally compact, second-countable topological abelian group, both with their Borel $\sigma$-algebras. Fix data $D$ consisting of additive automorphisms $\alpha_g$ of $A$ satisfying $\alpha_1=\operatorname{id}_A$ and $\alpha_{gh}=\alpha_g\circ\alpha_h$, with $(g,a)\mapsto\alpha_g(a)$ continuous; a measurable function $\beta:G\times G\to A$ satisfying $\alpha_g(\beta(h,k))+\beta(g,hk)=\beta(g,h)+\beta(gh,k)$ for all $g,h,k\in G$ and $\beta(1,g)=\beta(g,1)=0$ for all $g\in G$; and a specified open neighborhood $N$ of $1$ on whose square $\beta$ is continuous. For any extension points $x=(a,g)$ and $y=(b,h)$, where $a,b\in A$ and $g,h\in G$ are arbitrary, the defined operation returns the extension point

$$x\star y=\bigl(a+\alpha_g(b)+\beta(g,h),\;gh\bigr).$$

QAlgorithms.Papers.Independent.Intakethird100.P08.approximateIntegral

Let $G$ be a Hausdorff, locally compact, second countable topological group, and let $A$ be a Hausdorff, locally compact, second countable additive abelian topological group, both with their Borel measurable structures. Let $D$ specify additive automorphisms $\alpha_p$ with $\alpha_1=\mathrm{id}$, $\alpha_{pq}=\alpha_p\circ\alpha_q$, and jointly continuous evaluation; a measurable function $\kappa:G\times G\to A$ satisfying $\alpha_p(\kappa(q,r))+\kappa(p,qr)=\kappa(p,q)+\kappa(pq,r)$ and $\kappa(1,p)=\kappa(p,1)=0$; and an open neighborhood $U_0$ of $1$ such that $\kappa$ is continuous on $U_0^2$. Put $E=A\times G$ with the operation $(a,p)\star(b,q)=(a+\alpha_p(b)+\kappa(p,q),pq)$. For arbitrary functions $f,g:E\to\mathbb R$, the approximate integral $C(f,g)$ is the real infimum of all sums $\sum_{j=0}^{n-1}c_j$, where $n$ is any natural number, $c_j\ge0$, $u_j\in E$, and $f(s)\le\sum_{j=0}^{n-1}c_jg(u_j\star s)$ for every $s\in E$. The empty family $n=0$ has sum $0$ and is admissible exactly when $f\le0$ everywhere. Every admissible cost is nonnegative. If there are no admissible families, the real infimum used here has value $0$. There are no continuity, support, sign, or nonzero assumptions on $f$ or $g$.

QAlgorithms.Papers.Independent.Intakethird100.P08.continuousOneCocycles

Let $G$ be a topological group and $A$ a topological abelian group, and let $\rho$ specify additive automorphisms $\alpha_g$ of $A$ satisfying $\alpha_1=\operatorname{id}_A$ and $\alpha_{gh}=\alpha_g\circ\alpha_h$, with $(g,a)\mapsto\alpha_g(a)$ continuous. This declaration defines the set of all continuous functions $c:G\to A$ satisfying $c(st)=c(s)+\alpha_s(c(t))$ for every $s,t\in G$.

QAlgorithms.Papers.Independent.Intakethird100.P08.coverCosts

Let $G$ be a Hausdorff, locally compact, second countable topological group, and let $A$ be a Hausdorff, locally compact, second countable additive abelian topological group, both with their Borel measurable structures. Let $D$ consist of additive automorphisms $\alpha_p$, with $\alpha_1=\mathrm{id}$, $\alpha_{pq}=\alpha_p\circ\alpha_q$, and continuous evaluation $(p,a)\mapsto\alpha_p(a)$; a measurable function $\kappa:G\times G\to A$ satisfying $\alpha_p(\kappa(q,r))+\kappa(p,qr)=\kappa(p,q)+\kappa(pq,r)$ and $\kappa(1,p)=\kappa(p,1)=0$; and an open neighborhood $U_0$ of $1$ such that $\kappa$ is continuous on $U_0^2$. Put $E=A\times G$ and $(a,p)\star(b,q)=(a+\alpha_p(b)+\kappa(p,q),pq)$. For arbitrary functions $f,g:E\to\mathbb R$, the set of cover costs consists exactly of the real numbers $r$ for which there exist a natural number $n$, coefficients $c_j\in\mathbb R$, and points $u_j\in E$, indexed by $j=0,\ldots,n-1$, such that every $c_j\ge0$,

$$f(s)\le\sum_{j=0}^{n-1}c_jg(u_j\star s)\quad\text{for every }s\in E, \qquad r=\sum_{j=0}^{n-1}c_j.$$

No regularity or sign assumptions are imposed on $f$ or $g$. Zero coefficients are allowed. The choice $n=0$ is allowed: both sums then equal $0$, the coefficient and point families are empty, and this choice gives the cost $r=0$ exactly when $f(s)\le0$ for every $s$.

QAlgorithms.Papers.Independent.Intakethird100.P08.extensionBorelSpace

Let $G$ be a Hausdorff, locally compact, second-countable topological group, and let $A$ be a Hausdorff, locally compact, second-countable abelian topological group, both equipped with their Borel $\sigma$-algebras; either group may be trivial. Let $D=(\rho,c,N)$ consist of additive automorphisms $\rho_g:A\to A$ satisfying $\rho_1=\mathrm{id}$ and $\rho_{gh}=\rho_g\circ\rho_h$, with $(g,a)\mapsto\rho_g(a)$ continuous, a measurable map $c:G\times G\to A$ satisfying $\rho_g(c(h,k))+c(g,hk)=c(g,h)+c(gh,k)$ and $c(1,g)=c(g,1)=0$ for all $g,h,k\in G$, and an open set $N\subseteq G$ containing $1$ such that $c$ is continuous on $N\times N$. On $E_D=A\times G$, take the topology generated by the sets

$$\left\{\left(a+\rho_g(b)+c(g,h),\,gh\right): b\in U_A,\ h\in U_G\right\},$$

for all $(a,g)\in E_D$, open $U_A\subseteq A$ containing $0$, and open $U_G\subseteq G$ containing $1$ with $c$ continuous on $U_G\times U_G$; take its measurable structure to be the Borel $\sigma$-algebra of that topology. This instance certifies that the chosen measurable structure equals that Borel $\sigma$-algebra.

QAlgorithms.Papers.Independent.Intakethird100.P08.extensionCoordinates

Let $G$ be a Hausdorff, locally compact, second-countable topological group, and let $A$ be a Hausdorff, locally compact, second-countable abelian topological group, both equipped with their Borel $\sigma$-algebras; either group may be trivial. Let $D=(\rho,c,N)$ consist of additive automorphisms $\rho_g:A\to A$ satisfying $\rho_1=\mathrm{id}$ and $\rho_{gh}=\rho_g\circ\rho_h$, with $(g,a)\mapsto\rho_g(a)$ continuous, a measurable map $c:G\times G\to A$ satisfying $\rho_g(c(h,k))+c(g,hk)=c(g,h)+c(gh,k)$ and $c(1,g)=c(g,1)=0$ for all $g,h,k\in G$, and an open set $N\subseteq G$ containing $1$ such that $c$ is continuous on $N\times N$. For the associated set $E_D$ of points with an $A$-coordinate and a $G$-coordinate, the definition gives the function $q_D:E_D\to A\times G$ that sends each point with coordinates $a$ and $g$ to the ordered pair $(a,g)$.

QAlgorithms.Papers.Independent.Intakethird100.P08.extensionMeasurableSpace

Let $G$ be a Hausdorff, locally compact, second-countable topological group, and let $A$ be a Hausdorff, locally compact, second-countable abelian topological group, both equipped with their Borel $\sigma$-algebras; either group may be trivial. Let $D=(\rho,c,N)$ consist of additive automorphisms $\rho_g:A\to A$ satisfying $\rho_1=\mathrm{id}$ and $\rho_{gh}=\rho_g\circ\rho_h$, with $(g,a)\mapsto\rho_g(a)$ continuous, a measurable map $c:G\times G\to A$ satisfying $\rho_g(c(h,k))+c(g,hk)=c(g,h)+c(gh,k)$ and $c(1,g)=c(g,1)=0$ for all $g,h,k\in G$, and an open set $N\subseteq G$ containing $1$ such that $c$ is continuous on $N\times N$. Give $E_D=A\times G$ the topology generated by all sets

$$\left\{\left(a+\rho_g(b)+c(g,h),\,gh\right): b\in U_A,\ h\in U_G\right\},$$

where $(a,g)\in E_D$, $U_A\subseteq A$ is open and contains $0$, and $U_G\subseteq G$ is open, contains $1$, and has $c$ continuous on $U_G\times U_G$. The instance equips $E_D$ with the Borel $\sigma$-algebra of this topology, namely the smallest $\sigma$-algebra containing all its open sets.

QAlgorithms.Papers.Independent.Intakethird100.P08.extensionTopology

Let $G$ be a Hausdorff, locally compact, second-countable topological group, and let $A$ be a Hausdorff, locally compact, second-countable abelian topological group, both equipped with their Borel $\sigma$-algebras; either group may be trivial. Let $D=(\rho,c,N)$ consist of additive automorphisms $\rho_g:A\to A$ satisfying $\rho_1=\mathrm{id}$ and $\rho_{gh}=\rho_g\circ\rho_h$, with $(g,a)\mapsto\rho_g(a)$ continuous, a measurable map $c:G\times G\to A$ satisfying $\rho_g(c(h,k))+c(g,hk)=c(g,h)+c(gh,k)$ and $c(1,g)=c(g,1)=0$ for all $g,h,k\in G$, and an open set $N\subseteq G$ containing $1$ such that $c$ is continuous on $N\times N$. The instance equips the set of pairs $E_D=A\times G$ with the smallest topology in which every set

$$\left\{\left(a+\rho_g(b)+c(g,h),\,gh\right): b\in U_A,\ h\in U_G\right\}$$

is open, where $(a,g)$ ranges over all of $E_D$, $U_A$ ranges over open subsets of $A$ containing $0$, and $U_G$ ranges over open subsets of $G$ containing $1$ on whose square $c$ is continuous. These sets are the generators of the topology, so open sets are arbitrary unions of finite intersections of them, including the empty union and empty intersection.

QAlgorithms.Papers.Independent.Intakethird100.P08.identityPoint

Let $G$ be a Hausdorff, locally compact, second-countable topological group, and let $A$ be a Hausdorff, locally compact, second-countable abelian topological group, both equipped with their Borel $\sigma$-algebras; either group may be trivial. Let $D=(\rho,c,N)$ consist of additive automorphisms $\rho_g:A\to A$ satisfying $\rho_1=\mathrm{id}$ and $\rho_{gh}=\rho_g\circ\rho_h$, with $(g,a)\mapsto\rho_g(a)$ continuous, a measurable map $c:G\times G\to A$ satisfying $\rho_g(c(h,k))+c(g,hk)=c(g,h)+c(gh,k)$ and $c(1,g)=c(g,1)=0$ for all $g,h,k\in G$, and an open set $N\subseteq G$ containing $1$ such that $c$ is continuous on $N\times N$. In the associated set of pairs $E_D=A\times G$, the definition selects the point $\operatorname{identityPoint}_D=(0,1)$, where $0$ is the additive identity of $A$ and $1$ is the identity of $G$.

QAlgorithms.Papers.Independent.Intakethird100.P08.innerConjugation

Let $G$ be a Hausdorff, locally compact, second-countable topological group, and let $A$ be a Hausdorff, locally compact, second-countable abelian topological group, both equipped with their Borel $\sigma$-algebras; either group may be trivial. Let $D=(\rho,c,N)$ consist of additive automorphisms $\rho_g:A\to A$ satisfying $\rho_1=\mathrm{id}$ and $\rho_{gh}=\rho_g\circ\rho_h$, with $(g,a)\mapsto\rho_g(a)$ continuous, a measurable map $c:G\times G\to A$ satisfying $\rho_g(c(h,k))+c(g,hk)=c(g,h)+c(gh,k)$ and $c(1,g)=c(g,1)=0$ for all $g,h,k\in G$, and an open set $N\subseteq G$ containing $1$ such that $c$ is continuous on $N\times N$. For every $x=(a,g)$ and $y=(b,h)$ in the set $E_D=A\times G$, the definition assigns

$$\operatorname{innerConjugation}_D(x,y)= \left( a+\rho_g(b)+c(g,h) +\rho_{gh}\!\left(\rho_{g^{-1}}\bigl(-(a+c(g,g^{-1}))\bigr)\right) +c(gh,g^{-1}), \ ghg^{-1} \right).$$

QAlgorithms.Papers.Independent.Intakethird100.P08.kernelInclusion

Let $G$ be a Hausdorff, locally compact, second-countable topological group, and let $A$ be a Hausdorff, locally compact, second-countable abelian topological group, both equipped with their Borel $\sigma$-algebras; either group may be trivial. Let $D=(\rho,c,N)$ consist of additive automorphisms $\rho_g:A\to A$ satisfying $\rho_1=\mathrm{id}$ and $\rho_{gh}=\rho_g\circ\rho_h$, with $(g,a)\mapsto\rho_g(a)$ continuous, a measurable map $c:G\times G\to A$ satisfying $\rho_g(c(h,k))+c(g,hk)=c(g,h)+c(gh,k)$ and $c(1,g)=c(g,1)=0$ for all $g,h,k\in G$, and an open set $N\subseteq G$ containing $1$ such that $c$ is continuous on $N\times N$. Writing $E_D$ for the set of pairs $(a,g)\in A\times G$, the definition gives the function $i_D:A\to E_D$ by $i_D(a)=(a,1)$ for every $a\in A$.

QAlgorithms.Papers.Independent.Intakethird100.P08.leftTranslate

Let $G$ be a Hausdorff, locally compact, second countable topological group, and let $A$ be a Hausdorff, locally compact, second countable additive abelian topological group, both with their Borel measurable structures. Let $D$ specify additive automorphisms $\alpha_p$ satisfying $\alpha_1=\mathrm{id}$ and $\alpha_{pq}=\alpha_p\circ\alpha_q$, with $(p,a)\mapsto\alpha_p(a)$ continuous; a measurable function $\kappa:G\times G\to A$ satisfying $\alpha_p(\kappa(q,r))+\kappa(p,qr)=\kappa(p,q)+\kappa(pq,r)$ and $\kappa(1,p)=\kappa(p,1)=0$; and an open neighborhood $U_0$ of $1$ on whose square $\kappa$ is continuous. Elements of $E$ are pairs in $A\times G$. For any function $f:E\to\mathbb R$ and any $x=(a,p)\in E$, its left translate $L_xf:E\to\mathbb R$ is defined by

$$(L_xf)(b,q)=f\bigl(a+\alpha_p(b)+\kappa(p,q),\,pq\bigr).$$

No continuity, support, sign, or nonzero condition is imposed on $f$.

QAlgorithms.Papers.Independent.Intakethird100.P08.localConvolution

Let $G$ be a Hausdorff, locally compact, second countable topological group, and let $A$ be a Hausdorff, locally compact, second countable topological abelian group, both equipped with their Borel measurable structures. Let $D$ specify additive automorphisms $\rho_g$ of $A$ satisfying $\rho_1=\mathrm{id}$, $\rho_{gh}=\rho_g\circ\rho_h$, and joint continuity of $(g,a)\mapsto\rho_g(a)$; a measurable function $c:G\times G\to A$ satisfying $\rho_g(c(h,k))+c(g,hk)=c(g,h)+c(gh,k)$ and $c(1,g)=c(g,1)=0$; and an open neighborhood $N$ of $1$ on whose square $c$ is continuous. On $E=A\times G$, define $(a,g)\star(b,h)=(a+\rho_g(b)+c(g,h),gh)$. Equip $E$ with the topology generated by all sets $\{z\star(a,g):a\in P,\ g\in Q\}$, where $z\in E$, $P$ is an open neighborhood of $0$ in $A$, $Q$ is an open neighborhood of $1$ in $G$, and $c$ is continuous on $Q\times Q$, and use its Borel measurable structure. For any measure $\mu$ on $E$, any subset $M\subseteq E$, and any $x\in E$, this definition assigns the real number obtained from the extended nonnegative measure $\mu(M\cap\{x\star y:y\in M\})$ by retaining its value when it is finite and assigning $0$ when it is infinite. No measurability, finiteness, positivity, or invariance hypothesis is required here for $M$ or $\mu$; in particular, the defined function is identically zero when $M$ is empty.

QAlgorithms.Papers.Independent.Intakethird100.P08.locallyContinuousOneCocycles

Let $G$ be a topological group and $A$ a topological abelian group, with their given measurable structures, and let $\rho$ specify additive automorphisms $\alpha_g$ of $A$ satisfying $\alpha_1=\operatorname{id}_A$ and $\alpha_{gh}=\alpha_g\circ\alpha_h$, with $(g,a)\mapsto\alpha_g(a)$ continuous. This declaration defines the set of all functions $c:G\to A$ that are measurable, are continuous on some open set $U\subseteq G$ containing $1$, and satisfy $c(st)=c(s)+\alpha_s(c(t))$ for every $s,t\in G$.

QAlgorithms.Papers.Independent.Intakethird100.P08.normalizedIntegral

Let $G$ be a Hausdorff, locally compact, second countable topological group, and let $A$ be a Hausdorff, locally compact, second countable additive abelian topological group, both with their Borel measurable structures. Let $D$ consist of additive automorphisms $\alpha_p$, with $\alpha_1=\mathrm{id}$, $\alpha_{pq}=\alpha_p\circ\alpha_q$, and jointly continuous evaluation; a measurable function $\kappa:G\times G\to A$ satisfying $\alpha_p(\kappa(q,r))+\kappa(p,qr)=\kappa(p,q)+\kappa(pq,r)$ and $\kappa(1,p)=\kappa(p,1)=0$; and an open neighborhood $U_0$ of $1$ on whose square $\kappa$ is continuous. Put $E=A\times G$ and $(a,p)\star(b,q)=(a+\alpha_p(b)+\kappa(p,q),pq)$. For functions $h,k:E\to\mathbb R$, let $C(h,k)$ be the real infimum of the costs $\sum_{j=0}^{n-1}c_j$ over all finite families $c_j\ge0$, $u_j\in E$, satisfying $h(s)\le\sum_{j=0}^{n-1}c_jk(u_j\star s)$ for every $s$; $n=0$ is allowed, with empty sum $0$, and an empty set of admissible costs has infimum $0$. For arbitrary functions $g,\phi,f:E\to\mathbb R$, the normalized integral is defined by

$$J_{g,\phi}(f)=\frac{C(f,\phi)}{C(g,\phi)}.$$

No regularity, support, sign, or nonzero assumptions are imposed on these functions. Division is the total real division operation, so if $C(g,\phi)=0$, this definition gives $J_{g,\phi}(f)=0$.

QAlgorithms.Papers.Independent.Intakethird100.P08.quotientProjection

Let $G$ be a Hausdorff, locally compact, second-countable topological group, and let $A$ be a Hausdorff, locally compact, second-countable abelian topological group, both equipped with their Borel $\sigma$-algebras; either group may be trivial. Let $D=(\rho,c,N)$ consist of additive automorphisms $\rho_g:A\to A$ satisfying $\rho_1=\mathrm{id}$ and $\rho_{gh}=\rho_g\circ\rho_h$, with $(g,a)\mapsto\rho_g(a)$ continuous, a measurable map $c:G\times G\to A$ satisfying $\rho_g(c(h,k))+c(g,hk)=c(g,h)+c(gh,k)$ and $c(1,g)=c(g,1)=0$ for all $g,h,k\in G$, and an open set $N\subseteq G$ containing $1$ such that $c$ is continuous on $N\times N$. Writing $E_D$ for the set of pairs $(a,g)\in A\times G$, the definition gives the function $p_D:E_D\to G$ by $p_D(a,g)=g$ for every $(a,g)\in E_D$.

QAlgorithms.Papers.Independent.Intakethird100.P08.translatedRectangles

Let $G$ be a Hausdorff, locally compact, second-countable topological group, and let $A$ be a Hausdorff, locally compact, second-countable abelian topological group, both equipped with their Borel $\sigma$-algebras; either group may be trivial. Let $D=(\rho,c,N)$ consist of additive automorphisms $\rho_g:A\to A$ satisfying $\rho_1=\mathrm{id}$ and $\rho_{gh}=\rho_g\circ\rho_h$, with $(g,a)\mapsto\rho_g(a)$ continuous, a measurable map $c:G\times G\to A$ satisfying $\rho_g(c(h,k))+c(g,hk)=c(g,h)+c(gh,k)$ and $c(1,g)=c(g,1)=0$ for all $g,h,k\in G$, and an open set $N\subseteq G$ containing $1$ such that $c$ is continuous on $N\times N$. On the set $E_D=A\times G$, the definition specifies the family $\mathcal R_D$ of all subsets $V\subseteq E_D$ for which there exist a point $(a,g)\in E_D$, an open set $U_A\subseteq A$ containing $0$, and an open set $U_G\subseteq G$ containing $1$, such that $c$ is continuous on $U_G\times U_G$ and

$$V=\left\{\left(a+\rho_g(b)+c(g,h),\,gh\right): b\in U_A,\ h\in U_G\right\}.$$

There is no requirement that $U_G$ be contained in the specified set $N$.

QAlgorithms.Papers.Independent.Intakethird100.P08.twistedInv

Let $G$ be a Hausdorff, locally compact, second-countable topological group, and let $A$ be a Hausdorff, locally compact, second-countable abelian topological group, both equipped with their Borel $\sigma$-algebras; either group may be trivial. Let $D=(\rho,c,N)$ consist of additive automorphisms $\rho_g:A\to A$ satisfying $\rho_1=\mathrm{id}$ and $\rho_{gh}=\rho_g\circ\rho_h$, with $(g,a)\mapsto\rho_g(a)$ continuous, a measurable map $c:G\times G\to A$ satisfying

$$\rho_g(c(h,k))+c(g,hk)=c(g,h)+c(gh,k), \qquad c(1,g)=c(g,1)=0$$

for all $g,h,k\in G$, and an open set $N\subseteq G$ containing $1$ such that $c$ is continuous on $N\times N$. The points associated with $D$ are pairs $(a,g)\in A\times G$. For every such point, the definition assigns

$$\operatorname{twistedInv}_D(a,g) =\left(\rho_{g^{-1}}\bigl(-(a+c(g,g^{-1}))\bigr),\,g^{-1}\right).$$