Proposition 10

— If g is a $\mu$-measurable mapping with values in a topological space G (Hausdorff or not), there exists a $\mu$-crushing $(L_\beta)_{\beta \in B}$ of T such that the restrictions $g_{L_\beta}$ are continuous for all $\beta \in B$.

Consider a crushing $(K_\alpha)_{\alpha \in A}$ of T for $\mu$. Since the mapping $g$ is measurable, there exists for each $\alpha \in A$ a partition of $K_\alpha$ into a sequence $(K_{\alpha n})$ of compact sets and a locally negligible set $N_\alpha$, such that the restriction of $g$ to each of the sets $K_{\alpha n}$ is continuous. The family $(K_{\alpha n})_{(\alpha, n) \in A \times \mathbf{N}}$ is then the sought-for crushing. For, it is locally countable, and the set $N' = N \cup (\bigcup_\alpha N_\alpha)$ is locally negligible, because a compact set intersects at most a countable infinity of sets $N_\alpha$.