Proposition 9

If the function f is continuous on I × A, and if the integral $\int_I f(t, \alpha) dt$ is uniformly convergent on A, then the integral $\int_I dt \int_c^d f(t, \alpha) d\alpha$ is convergent, and one has

For every compact interval J contained in I, put $u_J(\alpha) = \int_J f(t, \alpha) dt$. The hypothesis entails that with respect to the filter of sections $\Phi$ of the directed set $\mathscr{K}(I)$ the continuous function $u_J$ converges uniformly on A to $\int_I f(t, \alpha) dt$; thus (II, p. 68, prop. 1), $\int_c^d d\alpha \int_J f(t, \alpha) dt$ has limit $\int_c^d d\alpha \int_I f(t, \alpha) dt$ with respect to $\Phi$; but, by prop. 8 (II, p. 77), one has

The preceding result thus means that the integral $\int_1 dt \int_c^{d} f(t, \alpha) d\alpha$ is convergent, and on passing to the limit with respect to $\Phi$ in the relation (18), one obtains (17).