Dominated convergence theorem for integrals depending on a parameter

Is there a theorem mirroring the dominated convergence theorem, but for integrals depending on a parameter instead?Meaning,

Let$$\begin{aligned}[t]f:I\times J&\longrightarrow \mathbb{R}\\(t,x)&\longmapsto f(t,x)\end{aligned}$$be a Riemann-integrable function with respect to $t$ on $I$, s.t. for some $a\in J, \lim_{x\to a} f(t,x)=F(t)$.

What other additional conditions are required in order to conclude that$$\lim_{x\to a} \int_I f(x,t)\,dt=\int_I F(t)\,dt$$?