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 --> E is a measurable mapping. Then there exists a sequence of functions in C-b( X) x 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.