Modification of almost surely integrable process is almost surely integrable?

Question

Let $f_t$, $g_t$ be two measurable stochastic processes, with $t\in [0,T]$, such that $g_s$ is a modification of $f_s$. Suppose $\mathbb{P}\{\int_0^T|f_s|ds<+\infty\}=1$. Is it true that $g_s$ is almost surely integrable?