Continuity of conditional measures associated to measure-preserving semiflows

Let $X$ be a standard probability space and $T_t$ a measure-preserving semiflow on $X$. We show that there exists a set $X_0$ of full measure in $X$ such that for any $x \in X_0$ and $t \geq 0$ there are measures $\mu_{x,t}^+$ and $\mu_{x,t}^-$ which for all but a countable number of $t$ give a distribution on the set of points $y$ such that $T_t(y) = T_t(x)$. These measures arise by taking weak$^*-$limits of suitable conditional expectations. Say that a point $x$ has a measurable orbit discontinuity at time $t_0$ if either $\mu_{x,t}^+$ or $\mu_{x,t}^-$ is weak$^*-$discontinuous in $t$ at $t_0$. We show that there exists an invariant set of full measure in $X$ such that any point in this set has at most countably many measurable orbit discontinuities. Furthermore we show that if $x$ has a measurable orbit discontinuity at time 0, then $x$ has an orbit discontinuity at time 0 in the sense of Orbit discontinuities and topological models for Bordel semiflows, D. McClendon.