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Let ${ ( X_t ) }_{ t \geq 0 }$ be an $\mathbb{R}^d$-valued stochastic process on $( \Omega, \mathcal{F}, P)$. I am trying to show that for any $A \in \mathscr{B} ( \mathbb{R}^d )$ the map $r : [ 0, \infty ] \times \mathbb{R}^d \rightarrow [ 0, 1 ]$, $( t, x ) \mapsto r ( t, x ) := P ( X_t + x \in A )$ is $\mathscr{B} ( [ 0 ,\infty ] ) \otimes \mathscr{B} ( \mathbb{R}^d )$-$\mathscr{B} ( [ 0, 1 ] )$-measurable. I tried to proceed in the following way; $$ r(t, x) = P ( X_t + x \in A ) = \int_{ \Omega } 1_{ \{ X_t + x \in A\}} P( d \omega) = \int_{ \Omega} 1_A \circ (X_t+x) P(dw). $$ I could further use the transformation rule by writing $$ \int_{ \Omega} 1_A \circ (X_t+x) P(dw) = \int_{ \mathbb{R^d}} 1_A \ P_{X_t +x}(dy). $$ However, I am not sure how to arrive at the meaurability arguemnt.

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You cannot do this without assuming that $(X_t)$ is a measurable process. (Measurable means $(t,\omega) \to X_t(\omega)$ is measurable). When this condition is satisfied you can see that $(t,x,\omega) \to I_A(X_t(\omega)+x)$ is measurable and the apply Fubini's Theorem to finish the proof.

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