Exercise 5, § 3

Let E be a set, $\mathscr{T}$ a tribe of subsets of E, m an abstract measure on $(E, \mathscr{T})$, and $p \geq 1$ a finite real number. For every numerical function f on E, one sets $N_p(f) = m^*(|f|^p)^{1/p}$ and one denotes by $\mathscr{F}^p$ the set of functions f such that $N_p(f)$ is finite (cf. §1, Exer. 4). One equips the vector space $\mathscr{F}^p$ with the semi-norm $N_p$, for which it is complete.

a) Let $\mathscr{N}$ be the set of functions f such that $m^*(|f|) = 0$, $\mathscr{V}$ the vector space generated by the finite functions belonging to $\mathscr{M}$, and $\mathscr{L}^p$ the smallest closed linear subspace of $\mathscr{F}^p$ containing the characteristic functions of the sets $A \in \mathscr{T}$ such that $m^*(A)$ is finite. Show that $\mathscr{L}^p = \mathscr{N} + (\mathscr{V} \cap \mathscr{F}^p)$.

b) Extend Lebesgue’s theorem (Ch. IV, §3, No. 7, Th. 6) to the present case, and examine in particular the case $p = 1$ (one will define the integral of a function $f \in \mathscr{L}^1$).

c) A subset $A$ of $E$ is said to be $m$-integrable if $\varphi_A \in \mathscr{L}^1$. Show that there then exists a set $A' \in \mathscr{T}$ such that $m^*(A \cap \complement A') = m^*(A' \cap \complement A) = 0$. Converse, in the case that $m(E)$ is finite.