Exercise 4, § 3

Let $\mu$ denote the Lebesgue measure on $\mathbf{R}$.

a) Let $\nu$ be a measure on $\mathbf{R}$ and $a\in \mathbf{R}$ such that $\nu (\{a\}) = 0$. For $x\in \mathbf{R}$, put

$\{\nu ([a, x])$ if $x\geqslant a$,

$\nu ([x, a])$ if $x\leqslant a$.

The function $f$ is continuous and the associated distribution satisfies $f'=\nu$.

b) Let $f$ be a distribution on $\mathbf{R}$ such that $f'$ is a measure $\nu$. The distribution $f$ is associated with the continuous function on $\mathbf{R}$ defined as in the preceding question.

c) Let $f$ be a distribution on $\mathbf{R}$ such that $f^{(k)}$ is a measure for every integer $k\geqslant 0$. Then $f$ is the distribution associated with a function in $\mathscr{C}^{\infty}(\mathbf{R})$.