Let $f:[a, b] \rightarrow \mathbb{R}$ be integrable with respect to Lebesgue measure $\mu$ on $[a, b]$. Prove that, if

$$\int_{J} f d \mu=0$$

for every sub-interval $J$ of $[a, b]$, then $f=0$ almost everywhere on $[a, b]$.

Now define

$$F(x)=\int_{a}^{x} f d \mu .$$

Prove that $F$ is continuous on $[a, b]$. Show that, if $F$ is zero on $[a, b]$, then $f$ is zero almost everywhere on $[a, b]$.

Suppose now that $f$ is bounded and Lebesgue integrable on $[a, b]$. By applying the Dominated Convergence Theorem to

$$F_{n}(x)=\frac{F\left(x+\frac{1}{n}\right)-F(x)}{\frac{1}{n}},$$

or otherwise, show that, if $F$ is differentiable on $[a, b]$, then $F^{\prime}=f$ almost everywhere on $[a, b]$.

The functions $f_{n}:[a, b] \rightarrow \mathbb{R}$ have the properties:

(a) $f_{n}$ converges pointwise to a differentiable function $g$ on $[a, b]$,

(b) each $f_{n}$ has a continuous derivative $f_{n}^{\prime}$ with $\left|f_{n}^{\prime}(x)\right| \leqslant 1$ on $[a, b]$,

(c) $f_{n}^{\prime}$ converges pointwise to some function $h$ on $[a, b]$.

Deduce that

$$h(x)=\lim _{n \rightarrow \infty}\left(\frac{d f_{n}(x)}{d x}\right)=\frac{d}{d x}\left(\lim _{n \rightarrow \infty} f_{n}(x)\right)=g^{\prime}(x)$$

almost everywhere on $[a, b]$.