Exercise 12, § 5

Let $\mu$ be a measure $\geqslant 0$ on $X$. Let $(f_n)$ be a sequence of integrable functions on $X$ that converges pointwise on $X$ to a function $f$.

a) Show that if $f$ is integrable and if

then for every $\varepsilon > 0$ there exist an integrable set $A$, an integrable function $g \geqslant 0$, and an integer $n_0$ such that, for every $n \geqslant n_0$,

and $|f_n(x)| \leqslant g(x)$ for all $x \in A$ (consider an integrable set $B$ such that

and such that $f$ is bounded on $B$, and apply Egoroff’s theorem).

b) Suppose that, for every $\varepsilon > 0$, there exist a measurable set $A$, an integrable function $g \geqslant 0$ and an integer $n_0$ such that, for every $n \geqslant n_0$, one has $\int |f_n| \varphi_{\mathbf{C}_A} d\mu \leqslant \varepsilon$ and $|f_n(x)| \leqslant g(x)$ for all $x \in A$. Show that, under these conditions, $f$ is integrable and $f_n$ tends to $f$ in the space $L^1$. Converse.

c) Suppose that $F = \mathbf{R}$; show by means of examples that the conditions of a) are not sufficient, and that the conditions of b) are not necessary, for $f$ to be integrable and for $\int f \, d\mu = \lim_{n \to \infty} \int f_n \, d\mu$ to hold.