Exercise 16, § 5

Let $X$ be a compact space, $\mu$ a positive measure on $X$. A bounded numerical function $f$ defined on $X$ is said to be continuous almost everywhere (for the measure $\mu$) in $X$ if the set of points of $X$ where $f$ is continuous (relative to $X$) has complement of measure zero.

a) Give an example of a function $f$ that is continuous almost everywhere and is such that there does not exist any continuous function $g$ equal almost everywhere to $f$.

b) Suppose that the support of $\mu$ is identical to $X$. Show that, for a bounded numerical function $f$ defined on $X$ to be equal almost everywhere to a function that is continuous almost everywhere in $X$, it is necessary and sufficient that there exist a subset $H$ of $X$, whose complement is negligible, such that the restriction $f|H$ of $f$ to $H$ is continuous (to prove that the condition is sufficient, note that $H$ is dense in $X$, and that the extension of $f|H$ to $X$ that is lower semi-continuous on $X$ (GT, IV, §6, Prop. 4) is a continuous function (relative to $X$) at every point of $H$). Deduce from this that $f$ is measurable.

c) Deduce from b) that if in addition $X$ is metrizable then, for every bounded numerical function $f$ that is equal almost everywhere to a function that is continuous almost everywhere in $X$, there exists a sequence $(f_n)$ of functions continuous on $X$, that is convergent at every point of $X$, and whose limit is almost everywhere equal to $f$ (cf. §4, Exer. 4 c)) (note that the proposition is true for a lower semi-continuous function $f$).

d) Let $A$ be a perfect set without interior point, contained in $X$, and of measure $> 0$ (cf. §4, Exer. 4 a)). Show that there does not exist any function that is continuous almost everywhere in $X$ and is equal almost everywhere to the upper semi-continuous function $\varphi_A$.