I am dealing with a stochastic integral (expected value) as follows: Suppose $c>0$, $X_t\ge0$ is an increasing process and $f\ge0$ is decreasing in $(-\infty,\infty)$. If $$ E\left(\int_{0}^{\infty}e^{-ct}|1-f(x+X_t)|dt\right)<\infty\;\;\mbox{for all}\;\;x, $$ does it imply that there exists $x_0\in\mathbb{R}$ such that $f(x_0)\le1$?
My argument is in the following: Note that $$ E\left(\int_{0}^{\infty}e^{-ct}|1-f(x+X_t)|dt\right)=\int_{0}^{\infty}\int_{0}^{\infty}e^{-ct}|1-f(x+y)|dtP(X_t\in dy), $$ which, by Fubini's theorem, equals to $$ \int_{0}^{\infty}e^{-ct}\int_{0}^{\infty}|1-f(x+y)|P(X_t\in dy)dt. $$ Since the integral is convergent, $\lim_{y\to\infty}|1-f(x+y)|=0$. Hence, $x_0$ exists.
Any comment, help or clarification would be very much appreciated. Thank you very much.