Integral over conditional expectation

Question

Suppose that $(t,x)\mapsto g(t, x)$ is continuous in $t\in [0,1]$ and $\mathsf{E}\sup_t |g(t,X)|<\infty$, where $X$ is some random variable living on $(\Omega,\mathcal{H},\mathsf{P})$. Is the following integral well-defined? $$ \int_0^1 \mathsf{E}[g(t,X)\mid \mathcal{F}]dt, $$ where $\mathcal{F}\subset\mathcal{H}$. If $Y(t,\omega)=\mathsf{E}[g(t,X)\mid \mathcal{F}](\omega)$, then $\{Y_t:0\le t\le 1 \}$ is a stochastic process. So I need to use Kolmogorov's continuity criterion to show that there exists an everywhere continuous modification $\tilde{Y}_t$. Then the integral under question is well defined if I replace $Y_t$ with $\tilde{Y}_t$. Is this reasoning correct?