Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 27G (Probability and Measure)

The monotone convergence theorem states that if
$0\le f_n\uparrow f$ pointwise, thenallowing the value $+\infty$.

Monotonicity of the integral shows that the increasing limit $L$ of the left side is at most $\int f$. Conversely, let $s$ be a nonnegative simple function with $s\le f$, and fix $0<c<1$. The setsincrease to the support of $s$. HenceLet $c\uparrow 1$ and take the supremum over all simple $s\le f$. This gives
$L\ge \int f$, proving equality.

Since $f_n\uparrow f$, the functionsare nonnegative and increase to $g=f-f_1$. The monotone convergence theorem gives
$\mu (g_n)\uparrow \mu (g)$. Because $f_1$ is integrable, adding the finite number
$\mu (f_1)$ yields

Integrability of $f_1$ is essential. On $\mathbb{R}$ with Lebesgue measure, letThen $f_n\uparrow 0$ pointwise, but every integral is $-\infty$, so
$\mu (f_n)$ does not increase to $\mu (0)=0$.