I'm having problems with this exercise. I've tried to apply the DOMINATED CONVERGENCE THEOREM but I couldn't. Could someone gives me any hint?
Suppose the function $x\rightarrow f(x,t)$ is $X$-measurable for each $t\in\mathbb{R}$, and the function $t\rightarrow f(x,t)$ is continuous on $\mathbb{R}$ for each $x\in X$. In addition, suppose that there are integrable functions $g$, $h$ on $X$ such that $|f(x,t)|\leqslant g(x)$ and such that the improper Riemann integral $$\int_{-\infty}^{+\infty}|f(x,t)|dt\leqslant h(x).$$ Show that $$\int_{-\infty}^{+\infty}\left[\int f(x,t)d\mu(x)\right]dt=\int\left[\int_{-\infty}^{+\infty}f(x,t)dt\right]d\mu(x),$$ where the integrals with respect to $t$ are improper Riemann integrals.
