Past exam of the options I course of the University of Oxford / 2007 / D (Integration)

The monotone convergence theorem states that if measurable nonnegative functions satisfy $f_n\uparrow f$ almost everywhere on a measure space, thenallowing either side to be $+\infty$.

Use the nonnegative increasing truncations $f_n=x^{-2}{1}_{[1/n,1]}$. They increase almost everywhere to the specified function. On each compact interval the continuous function has equal Riemann and Lebesgue integrals, so the fundamental theorem of calculus givesMonotone convergence yields $\int f=+\infty$. Since a nonnegative function is Lebesgue integrable exactly when its integral is finite, this function is not Lebesgue integrable.

The truncations $g_n=x^{1/2}{1}_{[1/n,1]}$ are measurable, nonnegative, and increase almost everywhere to $g$. As before their integrals agree with the corresponding Riemann integrals, givingMonotone convergence therefore gives the finite value $\boxed{\int g=2/3}$. Nonnegativity makes this the integral of the absolute value as well, proving Lebesgue integrability.

A dominated differentiation theorem is the following. Suppose $\varphi (x,t)$ is measurable in $x$ and differentiable in $t$ on an open interval, is integrable for at least one parameter value, and its parameter derivative is dominated in absolute value by an integrable function independent of $t$. Then its integral is differentiable andLocal versions require the domination only on a neighbourhood of each parameter. The proof bounds the difference quotients by the dominating function, using the fundamental theorem of calculus or the mean-value theorem, then applies dominated convergence. Apply the argument to real and imaginary parts for a complex integrand.

For fixed $t$, $f(x)e^{itx}$ is measurable and has absolute value $|f(x)|$, so is integrable. Its derivative with respect to $t$ is $ixf(x)e^{itx}$, whose absolute value is the assumed integrable function $|xf(x)|$. HenceDominated convergence also shows this derivative is continuous, though only differentiability is required.