Real Analysis - January 2026

Note: all statements require proofs. You can make references to standard theorems from the course, but you need to specify which exact standard fact you are referring to.

Problem 1

Show that the set of real numbers whose decimal expansion has digit 5 appearing infinitely often is a Borel set.

Problem 2

Suppose that $(X,S,\mu)$ is a measure space with $\mu(X)<+\infty$, and $f_1,f_2,\ldots$ is a sequence of measurable functions $f_j:X\to\mathbb R$ such that $f_j(x)\to+\infty$ for all $x\in X$. Show the following: for every $\varepsilon>0$ there exists $E\in S$ with $\mu(X\setminus E)<\varepsilon$, such that $f_j$ converges to $+\infty$ uniformly on $E$. The uniform convergence here means: for every $M>0$ there exists $N>0$ such that for all $j>N$ and $x\in E$, we have $f_j(x)>M$. Hint: this statement is similar to Egorov’s theorem.

Problem 3

Let $f:\mathbb R\to\mathbb R$ be an absolutely continuous function. Show the following: for every $\varepsilon>0$ there is $\delta>0$ such that for every Borel $E\subset\mathbb R$ with $\mu(E)<\delta$ we have $\mu(f(E))<\varepsilon$.

Problem 4

Let $(X,S,\mu)$ be a $\sigma$-finite measure space, and $f:X\to[0,+\infty)$ be a measurable function. Show:

$$\int_X f\,d\mu=2\int_{[0,+\infty)}\lambda\mu\{x\in X:f(x)>\lambda\}\,d\lambda.$$

Problem 5

Let $(X,S,\mu)$ be a measure space with $\mu(X)=1$. Show that $f\in L^\infty(X,\mu)$ if and only if $f\in L^p(X,\mu)$ for all $p\in[1,+\infty)$ and

$$\sup_p\|f\|_p<+\infty.$$