probability theory - Integrability of shifted random variable

Let $(\Omega,\mathcal{F},\mathbb{P})$ by a probability space, $X:\Omega\rightarrow[0,\infty)$ a (non-negative, finite) random variable and $F:[0,\infty)\rightarrow[0,\infty)$ a continuous function. Assume that

$$\mathbb{E}[\;F(X)\;]<\infty$$

I am wondering, if we can immediately draw the following conclusion:

$$\mathbb{E}[\;F(X + a)\;]<\infty$$

for all $a\in[0,\infty)$ (shift of the random variable).

The interesting case clearly is, if $X$ is unbounded, i.e. for every $M\in[0,\infty)$ there exists a set $A\in\mathcal{F}$ with $\mathbb{P}(A)>0$ and $X(\omega)>M$ for all $\omega\in A$).

Thank you very much for your help!

Answer

In general no: let $U$ be a random variable such that $U\geqslant e$ almost surely, $U$ is integrable but $U\notin\mathbb L^p$ for any $p\gt 1$. Then define $X=\log \log U$ and $F\colon x\mapsto \exp\left(\exp\left(x\right)\right)$. Then $F(X)=U$ and for any positive $a$, $F\left(X+a\right)=U^{e^a}$, which is not integrable.