Approximation of measurable mappings by sequences of continuous functions

Let $X$ be a completely regular Hausdorff space, $\mu$ a positive, finite Baire measure on $X$, and $E$ a separable metrizable locally convex space. Suppose $f: X \to E$ is a measurable mapping. Then there exists a sequence of functions in $C_{b}(X) \otimes E$ which converges to $f$ a.e. $[ \mu]$. If the function $f$ is assumed to be weakly continuous and the measure $\mu$is assumed to be $\tau$-smooth, then a separability condition is not needed.