Is this integral with condition expectation well-defined?

Question

Given a filtered probability space $(\Omega, \mathcal{F}, \{\mathcal{F}_t\}_{t \in [0, T]}, \mathbb{P})$, and a $\{\mathcal{F}_t\}$-adapted stochastic process $f(t, \omega)$ such that $$ \int_{0}^{T} f(t) dt < \infty. $$ The above integral is a Lebesgue integral.

Let $c$ be a constant with $0 < c < T$. Does the following integral still make sense? $$ \int_{0}^{T-c} \mathbb{E}[f(t + c)\vert \mathcal{F}_t] dt < \infty. $$

Answer 1

So as mentioned in the comments, if you can prove that

$E[f(\omega,t)]<\infty$,

then as mentioned here Does finite expectation imply bounded random variable?, we get $|f(\omega,t)|<\infty$ a.s.. Therefore, by Fubini and the tower property for

$$Y=\int E[f(\omega,t+c)\mid\mathcal{F}_{t}]dt$$

we have the condition

$$E[Y]=\int Ef(\omega,t+c)]dt<\infty$$

implies $|Y|<\infty$.