Below, $\mathcal{M}$ is the $\sigma$-algebra of Lebesgue measurable sets and $\lambda$ is Lebesgue measure.

(a) State the Lebesgue differentiation theorem for an integrable function $f: \mathbb{R}^{n} \rightarrow \mathbb{C}$. Let $g: \mathbb{R} \rightarrow \mathbb{C}$ be integrable and define $G: \mathbb{R} \rightarrow \mathbb{C}$ by $G(x):=\int_{[a, x]} g d \lambda$ for some $a \in \mathbb{R}$. Show that $G$ is differentiable $\lambda$-almost everywhere.

(b) Suppose $h: \mathbb{R} \rightarrow \mathbb{R}$ is strictly increasing, continuous, and maps sets of $\lambda$-measure zero to sets of $\lambda$-measure zero. Show that we can define a measure $\nu$ on $\mathcal{M}$ by setting $\nu(A):=\lambda(h(A))$ for $A \in \mathcal{M}$, and establish that $\nu \ll \lambda$. Deduce that $h$ is differentiable $\lambda$-almost everywhere. Does the result continue to hold if $h$ is assumed to be non-decreasing rather than strictly increasing?

[You may assume without proof that a strictly increasing, continuous, function $w: \mathbb{R} \rightarrow \mathbb{R}$ is injective, and $w^{-1}: w(\mathbb{R}) \rightarrow \mathbb{R}$ is continuous.]