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Let $(X,d)$ be a polish length space and $A_0,A_1 \subset X$ borel sets. Define the set of $t$-intermediate points as $$[A_0,A_1]_t = \{ \gamma(t) \; | \; \gamma \text{ is a constant speed geodesic s.t. } \gamma(i) \in A_i \; i = 0,1 \}. $$

Is $[A_0,A_1]_t$ borel measurable?

I am currently trying to pove that if $A_0,A_1$ are open then $[A_0,A_1]_t $ is open too. Perhaps one could then deduce that this holds true for all borel sets.

If $X$ is geodesic and the selection of geodesics is unique and depends continously on the boundary values then one can deduce that $[A_0,A_1]_t$ is analytic and therefore measurable w.r.t a probability measure $\mu$ on $X$.

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