Lebesgue Measure

Let $E_{k} \subset (0,1)(k=1,2, \ldots , N), \sum_{k=1}^{N} \lambda \left(E_{k}\right)>N-1$. Prove that

Let $E \subset S^{1}, z_{1}, \ldots , z_{N} \in S^{1}, \mu$ be Lebesgue measure on $S^{1}$ ("arc length"). Prove that if

then a suitably rotated set $E$ contains all the points $z_{1}, \ldots , z_{N}$, i.e., there exists a point $z_{0} \in S^{1}$ such that $z_{0} z_{k} \in E$ for $k=1, \ldots , N$.

What is the measure of the set of those points of the interval $(0,1)$ which, in their decimal expansion,

What is the measure of the set of numbers in the interval $(0,1)$ whose decimal representation contains the digit 0?

Let $E_{k} \subset (0,1)(k \in \mathbb {N})$. Is it true that there exists a subsequence $\left\{ E_{k_{j}}\right\}$ such that $\lambda \left(\bigcap_{j} E_{k_{j}}\right)>0$, if one of the following conditions holds:

Let $\theta_{k}>0, \sum \theta_{k}<+\infty$. Consider the set

Prove that $\lambda (A)=0$. In particular, if $r>2$, then for almost every $x \in \mathbb {R}$ there exists a number $c_{x}>0$ such that $\left|x-\frac{p}{q}\right| \geqslant \frac{c_{x}}{q^{r}}$ for all $p \in \mathbb {Z}$, $q \in \mathbb {N}$.

Let $0<\theta <1, E \subset \mathbb {R}^{m}, 0<\lambda_{m}(E)<\infty$. Prove that there exists a cube $\Delta$ such that $\theta \lambda_{m}(\Delta )<\lambda_{m}(E \cap \Delta )$.

Let $E \subset \mathbb {R}, E^{c}=\mathbb {R} \backslash E, E_{t}=\left\{ t+x \mid x \in E\right\}$ be the shift of the set $E$. Prove that if $\lambda (E)>0$ and $\lambda \left(E^{c}\right)>0$, then $\lambda \left(E_{t} \cap E^{c}\right)>0$ for some $t \in \mathbb {R}$.

Let $E, E_{0} \subset \mathbb {R}$ be arbitrary measurable sets of positive measure. Prove that

Let $E \subset \mathbb {R}, \lambda (E)>0$. Prove that if $\frac{1}{2}(x+y) \in E$ for any points $x, y \in E$, then $E$ is an interval (compare with problem I.1.21).

Find the Lebesgue measure of the orthogonal projection of the Cartesian square of the Cantor set onto an arbitrary line.

Let $v=\left\{ n_{k}\right\}$ be a strictly increasing sequence of natural numbers and $E_{\nu }=\left\{ \sum \varepsilon_{k} 2^{-n_{k}} \mid \varepsilon_{k}=0\right.$ or 1$\}$ (see problem I.1.34).

Let $E=\left\{ \left.\sum_{k \geqslant 1} \frac{\varepsilon_{k}}{k!} \right\rvert \, \varepsilon_{k}=0\right.$ or 1$\}$. Prove that $E$ is a homogeneous generalized Cantor set of measure zero (see the definition before problem I.1.29). What is its defining sequence $\left\{ l_{n}\right\}_{n \geqslant 0}$? Is it true that the measure of the set $E+E+\ldots +E$ ($m$ terms) equals zero for every $m \in \mathbb {N}$?

Prove that a generalized Cantor set with constant ratio (see the definition before problem I.1.29) has measure zero.

Construct a homogeneous generalized Cantor set $E \subset [0,1]$ of positive measure. Show that its measure can be arbitrarily close to one.

Prove the existence of a set $A \subset [0,1]$ such that $\lambda (A \cap \Delta )>0$ and $\lambda (\Delta \backslash A)>0$ for every (nonempty) interval $\Delta \subset (0,1)$.

Let $A \subset \mathbb {R}^{2}$ be a convex compact set, $A_{\varepsilon }$ the $\varepsilon$-neighborhood of the set $A$, i.e., the set $\bigcup_{x \in A} B(x, \varepsilon )$. Prove that $\lambda_{2}\left(A_{\varepsilon }\right)=a+b \varepsilon +c \varepsilon^{2}$, and find the coefficients $a, b, c$.

Let $A, B$ be convex compact sets in $\mathbb {R}^{3}$, with $A \subset B$. Prove that the surface area of the boundary of the set $A$ is no greater than the surface area of the boundary of the set $B$.

Prove that every (nonempty) open subset of the space $\mathbb {R}^{m}$ can be represented as the union of a sequence of pairwise disjoint balls and a set of measure zero.

Prove that the union of an arbitrary (even uncountable) family of nondegenerate segments is measurable.

Let $\left\{ E_{n}\right\}$ be a sequence of pairwise congruent subsets of the set $S^{1}$, satisfying the conditions (see problem I.1.20):

Prove that the sets $E_{n}$ cannot be measurable.

By a shift by $\theta$ modulo 1 we mean the map $x \mapsto \left\{ x+\theta \right\} , x \in [0,1)$, where $\left\{ y\right\}$ is the fractional part of the number $y$. Replacing the rotation of the circle by the angle $2 \pi \theta$ with the shift by $\theta$ modulo 1, construct, by analogy with problems I.1.20 and VIII.1.22, a nonmeasurable subset of the interval $[0,1)$.

Using the result of problem 1.23, show that a set $E \subset \mathbb {R}$ of positive measure contains a nonmeasurable subset.

Let $\mathfrak {A}$ be a system of subsets of the set $\mathbb {N}$ satisfying the conditions*): 1) if $\operatorname {card}(A)=\operatorname {card}(\mathbb {N} \backslash A)=\infty$, then $A \in \mathfrak {A}$ if and only if $\mathbb {N} \backslash \boldsymbol {A} \notin \mathfrak {A}$; 2) if $A \in \mathfrak {A}, C \subset \mathbb {N}, \operatorname {card}(C)<\infty$, then $A \cup C \in \mathfrak {A}$ and $A \backslash C \in \mathfrak {A}$.

Prove that the set $E=\left\{ \sum_{k \in A} 2^{-k} \mid A \in \mathfrak {A}\right\}$ is nonmeasurable.

We say that a set $E \subset \mathbb {R}^{m}$ generates a tiling if its shifts by all possible vectors with integer coordinates cover $\mathbb {R}^{m}$ and do not overlap one another, i.e.,

Prove that the measure of a measurable set generating a tiling equals one.

Let $E \subset \mathbb {R}^{m}, \lambda_{m}(E)>0, A$ be a countable dense subset of $\mathbb {R}^{m}$. Prove that $\lambda_{m}\left(\mathbb {R}^{m} \backslash \bigcup_{a \in A}(a+E)\right)=0$.

There is a sequence of $m$-dimensional balls whose radii tend to zero, while their total volume is infinite. Prove that finitely many of these balls can be placed in a cube so that they fill $99\%$ of its volume.

*) The existence of such a system of sets can be proved with the help of Zorn’s lemma, see, for example, p. 55 in [Ke].

Prove that if pairwise disjoint disks contained in a square cover it up to a set of measure zero, then the sum of the lengths of the boundary circles is infinite.