Exercise 8, § 3

Let I and J be two arbitrary intervals in $\mathbf{R}$, and $\mathbf{f}$ a function defined and continuous on $I \times J$, with values in a complete normed space E over $\mathbf{R}$. Suppose that:

1 the integral $\int_I \mathbf{f}(x, y) dx$ is uniformly convergent when y runs through an arbitrary compact interval contained in J;

2 the integral $\int_J \mathbf{f}(x, y) dy$ is uniformly convergent when x runs through an arbitrary compact interval contained in I;

3 if, for every compact interval H contained in I, one puts $u_H(y) = \int_H \mathbf{f}(x, y) dx$, the integral $\int_J u_H(y) dy$ is uniformly convergent for $H \in \mathfrak{K}(I)$ (the right directed ordered set of compact intervals contained in I).

Under these conditions, show that the integrals $\int_I dx \int_J \mathbf{f}(x, y) dy$ and $\int_J dy \int_I \mathbf{f}(x, y) dx$ exist and are equal.