Let $X$ be a continuous stochastic process 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.?