Exercise 4, § 4

Let $\mu$ be Lebesgue measure on the interval $X = [0, 1]$.

a) Define in $X$ a nowhere dense perfect set $A$ whose measure is an arbitrary number $\alpha$ such that $0 \leq \alpha < 1$ (use the method of construction of Cantor’s triadic set).

b) Define in $X$ a sequence $(A_n)$ of pairwise disjoint nowhere dense integrable sets, such that $\mu(A_n) = 2^{-n}$ and such that every interval contiguous to $B_n = \bigcup_{k=1}^n A_k$ contains a subset of $A_{n+1}$ of measure $> 0$. If $A = \bigcup_n A_n$, show that $A$ is a meager set of measure 1, and $\complement A$ a negligible non-meager set.

c) Let $H = \bigcup_{n=0}^\infty A_{2n+1}$; show that, for every open interval $I \subset X$, the intersections of $I$ with $H$ and with $\complement H$ have measure $> 0$. Let $f$ be a function equal almost everywhere to the characteristic function $\varphi_H$; show that there does not exist any sequence $(f_n)$ of continuous functions on $X$, convergent at every point of $X$, whose limit is equal to $f$ (note that $f$ is necessarily discontinuous at every point of $X$, and make use of Exer. 22 of GT, IX, §5).