Let $p(r,x):=(4 \pi r)^{-d/2}e^{-\frac{|x|^2}{4r}},r>0,x \in \mathbb{R}^d.$ Let $\mathcal{P}$ be the predictable $\sigma$-algebra (on $\mathbb{R}_+ \times \Omega$ generated by $\{0\}\times F_0$ and $]r,u] \times F$ where $0 \leq r<u,F_0 \in \mathcal{F}_0,F \in \mathcal{F}_r$).
Let $(X_r)_{r \in \mathbb{R}_+}$ be a predictable stochastic processes such that $\sup_{r\in \mathbb{R}_+}e^{-r}E[X_r^2]<\infty,$ $$\forall u \in \mathbb{R}_+,x \in \mathbb{R}^d,E\left[\int_0^u(p(u-r,x)f(X_r))^2dr\right]<\infty,$$ where $f:\mathbb{R} \to \mathbb{R}$ is a Lipschitz function.
Prove that $(u,\omega,x) \to \int_0^u p(u-r,x)f(X_r(\omega))dB_r(\omega)$ is $\mathcal{P} \otimes \mathcal{B}(\mathbb{R}^d)$-measurable.
From stochastic integrals property we know that for all $x \in \mathbb{R}^d,(u,\omega) \to \int_0^u p(u-r,x)f(X_r(\omega))dB_r(\omega)$ is $\mathcal{P}$-measurable, I tried proving that for all $(u,\omega) \in \mathbb{R}_+ \times \Omega, x \to \int_0^u p(u-r,x)f(X_r(\omega))dB_r(\omega)$ is continuous but it seems difficult.
Any ideas how to prove it ? Is it possible to do it using an approximation argument?