Exercise 4, § 5

Let $\mu$ be Lebesgue measure on $X = [0, 1[$, and let $(H_n)$ be a partition of X into an infinite sequence of sets having the power of the continuum none of which is measurable, and which are such that for every union H of a finite number of sets $H_n$, $\mu_*(H) = 0$ (§4, Exer. 8). Let $\sigma_n$ be a bijection of the interval $]1/(n+1), 1/n]$ onto $H_n$. For every number y such that $0 < y \leq 1$, let n be the integer such that $1/(n+1) < y \leq 1/n$. Define $f_y$ to be the characteristic function of the set reduced to the point $\sigma_n(y)$; for every $x \in X$, $f_y(x)$ tends to 0 as y tends to 0. Show that there does not exist any compact set $K \subset X$ of measure $> 0$ such that $f_y$ tends uniformly to 0 on K.