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.
