using dominated convergence theorem in swapping limit

Question

Let $(X, \mathscr{A}, \mu)$ be a measure space, let $g$ be a $[0,+\infty]$-valued integrable function on $X$, and let $f$ and $f_t$ (for $t$ in $[0,+\infty)$ ) be real-valued $\mathscr{A}$-measurable functions on $X$ such that $$ f(x)=\lim _{t \rightarrow+\infty} f_t(x) $$ and $$ \left|f_t(x)\right| \leq g(x) \text { for } t \text { in }[0,+\infty) $$ hold at almost every $x$ in $X$. Show that $\int f d \mu=\lim _{t \rightarrow+\infty} \int f_t d \mu$.
here condition above implies, $\left|f_t(x)\right|$ is integrable, but beside that I am not able to use other condition to prove result.