# Stochastic processes and continuity of expectation

Let $$X$$ be a stochastic process with a.s. continuous sample paths on $$[0, 1]$$ such that $$\mathbb E [X_t]$$ is finite for all $$t \in [0, 1]$$. Given any non null subset $$Y$$ of the probability space, define $$\mathbb Q_Y$$ to be the restricted probability measure $$\mathbb Q_Y [E] = P(E \cap Y)/P(Y)$$.

Does it follow there exists some non null $$Y$$ such that that the function $$f: [0, 1] \to R$$ defined $$f(t)$$ $$=$$ $$\mathbb E_{Q_Y} [X_t]$$ is continuous a.e.?

Assuming you don't mean continuous sample paths, the answer is no. For example, $$X_t =$$ the indicator of the rationals with probability 1, or on a space that is an atom of mass 1 if you prefer. Then there is only 1 set of positive probability and it doesn't work.
Since you do mean continuous sample path, the answer is 'yes', take $$Y = \lbrace max |X_t| < A \rbrace$$ which can be made to be of positive probability by the continuity of $$X_t$$. Then, by dominated convergence (convergence because of continuity, and dominated by the bound A) $$f$$ is continuous.