# Show the map is measurable w.r.t. the product space

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.

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.