(1) Suppose a function $f$ has a [Lebesgue] measurable domain and is continuous except at a finite number of points. Is $f$ is necessarily [Lebesgue] measurable?
Comments For (1), If $f$ is defined on a [Lebesgue] measurable set $E$ and is continuous except for a finite number of values say $x_1,x_2, … x_n$ are those points of discontinuity, then can't we describe the pre-images of $f$ just as a finite union of of the pre-images of the collection of continuous functions $\{f_i\}_{i=1}^{n}$, where each $f_i$ is defined up between each point of discontinuity of $f$? Or am I missing something here?
(2) Suppose the function $f$ is defined on a measurable set $E$ and has the property that $\{x \in E | f(x) > c\}$ is measurable for each rational number $c$. Is $f$ necessarily [Lebesgue] measurable?
Comments I got this one, thanks to everyone who commented.
Any hints would be appreciated.
The text being used is Royden-Fitzpatrick 4th Edition.
Best Answer
What about this for the first question:
Let $A$ be the measurable domain of $f$ and let $X=\{x_i\}_{i=1}^n$ be the finite collection of discontinuities. Since $|X|$ is finite, we can order them from smallest, say $x_1$, to largest, say $x_n$. Then for $1\le i\le n+1$ define $$ A_i=A\cap (x_{i-1},x_i), $$ where $x_0=-\infty$ and $x_{n+1}=\infty$. Observe that each $A_i$ is measurable and $$ \{x\in A\,:\, f(x)>c\}=\left(\bigcup_{i=1}^{n+1}\{x\in A_i\,:\, f(x)>c\}\right) \cup \{x\in X\,:\,f(x)>c\} $$ For each $c$, each set in the indexed union is measurable (because $f$ is measurable on each A_i) and each subset of $X$ is either empty or contains no more than $n$ points and is therefore measurable as well.