Real Analysis - January 2025

Note: all statements require proofs. You can make references to standard theorems from the course; however, you need state the relevant part of the theorem in your own words, unless it’s a well known named theorem. For example, ”we had a theorem in the class that said that any continuous function on a compact subset of $\mathbb{R}^n$ is uniformly continuous” is a good reference, and ”by the uniqueness theorem from the class, $f$ is unique” is not a good reference.

Problem 1

Let $f_n:[0,1]\to[0,+\infty)$ be a sequence of measurable functions, such that

$$\lim_{n\to+\infty}\int_0^1 f_n(x)\,dx=0.$$

Show that there is a subsequence $f_{n_k}$ that converges to 0 almost everywhere as $k\to+\infty$. Explain why this does not necessarily hold for the original sequence (by providing a counterexample).

Problem 2

Let $(X,S,\mu)$ be a measure space with $\mu(X)<+\infty$. Show that $L^p(X,\mu)\subset L^q(x,\mu)$ for $1\le p\le q\le+\infty$.

Problem 3

Show that the set of points of every smooth curve in $\mathbb R^2$ has Lebesgue measure zero. More precisely, let $f,g:[0,1]\to\mathbb R$ be two continuously differentiable functions (that is, $f,g,f\prime,g\prime$ are continuous on $[0,1]$). Show that $\{(f(t),g(t)):t\in[0,1]\}$ has zero Lebesgue measure in $\mathbb R^2$.

Problem 4

Let $(X,S,\mu)$ be a measure space with $\mu(X)=1$ and $h\in L^1(X,\mu)$. Prove the following inequality for all $c>0$:

$$\mu\left(\left\{x\in X:h(x)-\int_Xh\,d\mu\ge c\right\}\right)\le\frac1{c^2}\left(\int_X h^2\,d\mu-\left(\int_X h\,d\mu\right)^2\right).$$

Problem 5

Let $(X,S,\mu)$ be a measure space with $\mu(X)<+\infty$. Suppose that, for every $B\in S$ with $\mu(B)>0$, there exists $A\subset B$, $A\in S$, with $0<\mu(A)<\mu(B)$. Show that $\mu(E)$ attains arbitrarily small positive values. More precisely: show that, for every $\varepsilon>0$, there exists $E\in S$ such that $0<\mu(E)<\varepsilon$.