In Baldi's book, Stochastic Calculus, Proposition 2.1 states that a right continuous adapted process, taking values in a topological space $E$ will be progressively measurable.
The proof starts out in the standard way, producing a sequence of elementary processes, $X^{(n)}$, with pointwise convergence to $X$ which are progressively measurable. It concludes that as the limit of such processes, it is also progressively measurable.
But is it true that this should hold in a topological space? Results that I am aware of on convergence of measurable functions to a measurable function are all on spaces which are at least metrizable. In fact, there appear to be counterexamples when you do not have a metric.
Does the additional structure here (right continuity? adaptedness?) allow us to circumvent having a metric?
Or am I just parsing the book incorrectly?
If the topological space $E$ is a separable metric space, then the Borel $\sigma$-field is generated by the open balls, and Baldi's assertion is correct.
Also, the discussion here: When are pointwise limits of measurable functions measurable? is instructive. [N.B.: By a theorem of Tikhonov, a second countable regular space is metrizable.]