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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.

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  • $\begingroup$ Have you studied Fubini's or Tonelli's theorems? $\endgroup$ – P. J. Feb 26 at 5:40
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    $\begingroup$ Can you show how you tried to apply the DCT? Otherwise this type of problem statement question typically gets closed. This point of this problem which appears before any reference to Fubini's theorem is to prove that the Lebesgue and improper Riemann integrals iterate from first principles under these conditions. Improve the question and I can point you in the right direction. $\endgroup$ – RRL Feb 26 at 21:16

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