Lemma 4

Let Y and Z be locally compact topological spaces, $f: X\rightarrow Y$ and $g: Y\rightarrow Z$ universally measurable mappings. The mapping $g\circ f: X\rightarrow Z$ is universally measurable.

Let $\mu$ be a positive measure with compact support on X and let C be its support. The restriction of $f$ to C is $(\mu|$C)-proper. The mapping $g$ is measurable with respect to the image measure $(f|C)(\mu|C)$, hence the mapping $(g\circ f)|C =g\circ (f|C)$ is $(\mu|$C)-measurable. Consequently, the mapping $g\circ f$ is $\mu$-measurable. The lemma follows.

We denote by $\mathscr{L}_u(X)$ the vector space of complex-valued functions which are universally measurable on X, and by $\mathscr{L}_u^{\infty}(X)$ the subspace of functions $f\in \mathscr{L}_u(X)$ which are bounded on X. For $f\in \mathscr{L}_u^{\infty}$(X), we put $\|f\|_{\infty}=$ sup$_{x\in X}|f(x)|$.