I am reading "Analysis on Manifolds" by James R. Munkres.
On p.111, there is the following exercise(exercise 2) for section 13("The Integral over a Bounded Set").
Let $A$ be a rectangle in $\mathbb{R}^k$; let $B$ be a rectangle in $\mathbb{R}^n$; let $Q=A\times B$. Let $f:Q\to\mathbb{R}$ be a bounded function. Show that if $\int_Q f$ exists, then $$\int_{\mathbf{y}\in B}f(\mathbf{x},\mathbf{y})$$ exists for $\mathbf{x}\in A-D$, where $D$ is a set of measure zero in $\mathbb{R}^k$.
I have no idea.
The following propositions are all the propositions in section 13. (I added exercise 1 in this picture.)
I guess I must use one of these propositions or exercise 1 to solve this exercise.
But I don't know which proposition I must use.
RRL, Thank you very much for your elegant answer. RRL used Theorem 12.2(Fubini's theorem) on p.100 in Section 12 and Theorem 11.3 on p.96 in Section 11 and Definition(integrable over Q) on pp.84-85 in Section 10.
RRL didn't use any proposition in Section 13.
I wonder why Munkres set Exercise 2 for Section 13 and did not set Exercise 2 for Section 12.
I noticed that RRL used Theorem 13.3(a) on p.106 in Section 13.
So Munkres set Exercise 2 for Section 13.
I like Munkres.
Best Answer
Since $f$ is bounded, the upper and lower integrals $\overline{\int_B} f(x,y) \, dy,\,\,\,\underline{\int}_B f(x,y) \, dy,$ exist for each $x \in A$. If $f$ is Riemann integrable on $Q$, then by Fubini's theorem
$$\int_A \left(\overline{\int_B} f(x,y) \, dy \right) \, dx = \int_A \left(\underline{\int}_B f(x,y) \, dy \right) \, dx,$$
and, hence,
$$\int_A \left(\overline{\int_B} f(x,y) \, dy - \underline{\int}_B f(x,y) \, dy \right) \, dx = 0.$$
Since the difference of upper and lower Darboux integrals is nonnegative, it follows that
$$\overline{\int}_B f(x,y) \, dy = \underline{\int}_B f(x,y) \, dy ,$$
except possibly on a set $D \subset A$ of measure zero, and the function $f(x, \cdot) :y \mapsto f(x,y)$ is Riemann integrable for every $x \in A-D$.