Equivalence of the two distributional Sobolev membership presentations

For every subset $\Omega\subseteq\mathbb R^n$, every order $k\ge0$, exponent $p\in[0,\infty]$, and real function $f$, the Compactness and Sobolev chapter predicates for $f\in W^{k,p}(\Omega)$ are equivalent. No openness or positive dimension assumption is needed.

The Compactness definition quantifies over a finite set of multi-indices and uses a flattened word of coordinate derivatives. The Sobolev definition quantifies over all multi-indices of total order at most $k$, iterates each coordinate derivative, and separately asks that $f$ be locally integrable. The zero multi-index supplies this local integrability in the first presentation. Flattening the derivative word and bounding each coordinate of a multi-index by its total degree identify the remaining clauses.