Summable Functions

Let the union of any three sets of the family $E_{1}, \ldots , E_{N}$ coincide with the interval $\left[0,1\right]$. Set

Express in terms of $S_{1}$ and $S_{2}$ the integral $\mathcal{Y}=\int_{0}^{1}\left(\sum_{k=1}^{N} \chi_{E_{k}}(x)\right)^{3} d x$.

Let each point of the interval $\left[0,1\right]$ belong to at least $k$ of the sets $E_{1}, \ldots , E_{N}$. Prove that for some $n$ the inequality $\lambda \left(E_{n}\right) \geqslant \frac{k}{N}$ holds.

Let $E \subset \mathbb {R}^{m}, \lambda_{m}(E)<\infty , f \in \mathscr {L}^{0}(E)$. Prove the equivalence of the following conditions:

1. $f \in \mathscr {L}(E)$;

2. $\sum_{k \geqslant 1} \lambda_{m}(E(\left|f\right| \geqslant k))<\infty$;

3. $\sum_{k \geqslant 1} k \lambda_{m}(E(k \leqslant \left|f\right|<k+1))<\infty$.

Let the sets $E_{k} \subset \mathbb {R}^{m}(k \in \mathbb {N})$ satisfy the condition $\sum_{k \geqslant 1} \lambda_{m}\left(E_{k}\right)<\infty$, and let

Prove that the sets $A_{n}, B_{n}$ are measurable and

Let $x_{n}>0, x_{n} \rightarrow 0, f(t)=\operatorname {card}\left\{ n \in \mathbb {N} \mid x_{n}>t\right\}$. Prove that $\int_{0}^{+\infty } f(t) d t=\sum x_{n}$.

Can an everywhere differentiable function on $\left[0,1\right]$ have a derivative that is not summable on $\left[0,1\right]$?

For which $p \in \mathbb {R}$ are the following functions summable on the half-line ($0,+\infty$)

Are the functions $f$ indicated below summable on the square $\left[-1,1\right] \times [-1,1]$? Find the iterated integrals

if:

For which $p \in \mathbb {R}$ is the function $f(x, y)=\left|1-x y\right|^{-p}$ summable on the set $E$ in the following cases:

Let $p, t>0, E \subset \mathbb {R}^{m}, f \in \mathscr {L}^{0}(E)$. Prove that

In particular, for $p=2$ we obtain Chebyshev's inequality:

Prove that if $\left|f\right|^{p} \in \mathscr {L}(E)\left(p>0, E \subset \mathbb {R}^{m}\right)$, then

Give an example showing that the converse statement is false.

Let $p>0, E \subset \mathbb {R}^{m}, f \in \mathscr {L}^{0}(E)$ and $\lambda_{m}(E(\left|f\right| \geqslant t))=O\left(t^{-p}\right)$ as $t \rightarrow +\infty$. Prove that if $\lambda_{m}(E)<\infty$, then $\left|f\right|^{p-\varepsilon } \in \mathscr {L}(E)$ for any $\varepsilon \in (0, p)$.

Let $E \subset \mathbb {R}^{m}, f \in \mathscr {L}(E)$. Prove that

Let $f, g \in \mathscr {L}^{0}(E), f, g \geqslant 0$. Prove that if $\int_{E} g(x) d x=1$ and $\int_{E} g(x) f(x) d x<\infty$, then

Can the condition $\int_{E} g(x) f(x) d x<\infty$ be dropped?

Let $f$ be a measurable nonnegative function on $\mathbb {R}$. Prove that for every $\varepsilon >0$ the following inequalities hold:

Let $f$ be a summable nonnegative function on $\mathbb {R}$, $\left\{ \left(a_{n}, b_{n}\right)\right\}_{n \in \mathbb {Z}}$ a two-sided sequence of intervals. Suppose there exists $\delta >0$ such that $a_{n+1} \geqslant b_{n}+\delta$ for all $n \in \mathbb {Z}$. Prove that then

Let $x \in \mathbb {R}, x=\left[x\right]+\sum \varepsilon_{k} 2^{-k}$ be the binary representation of the number $x$.

Let $x \in \mathbb {R}, x=\left[x\right]+\sum \varepsilon_{k} \cdot 10^{-k}$ be the decimal representation of the number $x$. Let $j$ be one of the digits $0,1, \ldots , 9$, and let

Find the integrals ($k, m \in \mathbb {N}, k \neq m$)

Let $a \in \mathbb {R}, a \neq 0, a \neq 1, x=\sum_{k \geqslant 1} \varepsilon_{k}(x) \cdot 10^{-k}$ be the decimal representation of the number $x, x \in (0,1)$. Set

Prove that the function $f$ is measurable, and find $\int_{0}^{1} f(x) d x$.

In constructing the Cantor set from the interval $\left[0,1\right]$, intervals of length $\frac{1}{3}, \frac{1}{9}$, and so on were successively removed (see the text before problem I.1.29). Let $f(x)=n$ on intervals of length $3^{-n}$, $f(x)=0$ on the Cantor set.

Let $K$ be a closed subset of the interval $\left[a, b\right]$,

the distance from the point $y \in \mathbb {R}$ to the set $K$. Prove that for any $s>0$ the function

is summable on $K$.

Prove that for a nonnegative function $f$ the equalities hold $(0<a \leqslant \infty )$

Compute the integral ($p>-1$)

Prove by induction the validity of the equalities ($m \geq 2$)

Let the function $f$, continuous on $\mathbb {R}^{m}$, satisfy the condition $f(t x)=t f(x)>0$ for $x \neq 0, t>0$, and let $A$ be a nondegenerate linear transformation on $\mathbb {R}^{m}$. Prove that

Let $E$ be a measurable subset of the half-line $(0,+\infty )$, $A_{E}=\left\{ x \in \mathbb {R}^{m} \mid \left\| x\right\| \in E\right\}$. Prove that

What is the "Gaussian volume" of the ball $B(0, r)$?

Let $a>0$, and let $K_{a}$ be the cone in the space $\mathbb {R}^{4}$:

Find $\gamma_{4}\left(K_{a}\right)$.

Compute the volume of the generalized torus $T \subset \mathbb {R}^{4}$

where $B$ is the disk with center at the point $(0, a)$ and radius $R \leqslant a$.

Let $A \subset \mathbb {R}^{2}$ be a compact set lying in the upper half-plane. If $T_{0}$ is the solid in $\mathbb {R}^{3}$ obtained by rotating the set $A$ about the axis $x_{1}$, then by Guldin's theorem $\lambda_{3}\left(T_{0}\right)=\lambda_{2}(A) 2 \pi y_{c}$, where $y_{c}$ is the ordinate of the centroid of the set $A$. Regarding $\mathbb {R}^{2}$ as canonically embedded in $\mathbb {R}^{4}$, consider the solid $T$ in $\mathbb {R}^{4}$ "obtained by rotating $A$ about the axis $x_{1}$":

Prove that the following modification of Guldin's theorem holds:

where $\rho$ is the "radius of gyration" of the set $A$ with respect to the axis $x_{1}$, i.e., the positive number defined by the equality

Let $A \subset \mathbb {R}^{2}$ be a compact set lying in the upper half-plane. Regarding $\mathbb {R}^{2}$ as canonically embedded in $\mathbb {R}^{m}$, consider in $\mathbb {R}^{m}$ the set $T$, the "solid obtained by rotating $A$ about the axis $x_{1}$":

Prove that

Compute the integral

Find the Lebesgue measure of the set

Let $F \in C(\mathbb {R}), a \in \mathbb {R}^{m}$. Reduce the integrals of $F((x, a))$ over the unit ball and over the unit sphere in $\mathbb {R}^{m}$ to integrals over the segment $\left[-1,1\right]$.

Let $a \in \mathbb {R}^{m}, A: \mathbb {R}^{m} \rightarrow \mathbb {R}^{m}$ be a nondegenerate linear transformation, $F \in C(\mathbb {R})$. Prove that

where $c=\left\| \left(A^{-1}\right)^{*}(a)\right\| , \sigma_{m-1}=\mu_{m-1}\left(S^{m-1}\right)$.

Let $\theta$ be the angle between the nonzero vectors $a, b \in \mathbb {R}^{m} (0 \leqslant \theta \leqslant \pi )$. Prove that

Let $I_{m}=\int_{0}^{\infty } \gamma_{m}(B(0, r))\left(1-\gamma_{m}(B(0, r))\right) d r$. Prove that

Let $A \subset \mathbb {R}^{m}$ be an open set that is star-shaped with respect to the origin,

the gauge function of the set $A$, and let $\mu$ be a probability measure on $\mathbb {R}^{m}$ such that $\int_{\mathbb {R}^{m}}\left|(x, y)\right| d \mu (x)<+\infty$ for every $y \in \mathbb {R}^{m}$.

Prove that

Let the function $f$ be summable on the cube $Q=[0,1)^{m}$ and have period 1 in each variable. Prove that

Let $T$ be an arbitrary convex compact body in $\mathbb {R}^{m}$, $V=\lambda_{m}(T)$. Prove that if the $(m-1)$-dimensional volume of the projection of $T$ onto any hyperplane is not less than $S$, then $\operatorname {diam} T \leqslant m V / S$.

Let $T \subset \mathbb {R}^{m}$ be a convex compact body that is centrally symmetric with respect to the origin, and let $\mathscr {E}$ be an ellipsoid of maximal volume contained in $T$:

Prove that $T \subset \sqrt{m} \mathscr {E}$.

Let $K \subset \mathbb {C}$ be a closed bounded set of positive (planar) measure. Define the function $\varphi$ by the equality

Prove that

Identifying $\mathbb {C}^{2}$ with $\mathbb {R}^{4}$ in the natural way, consider on the sphere $S^{3}$ the functions $\varphi_{m, n}(m, n=0,1,2, \ldots )$, defined by the equalities $\varphi_{m, n}\left(\xi_{1}, \xi_{2}\right)=\xi_{1}^{m} \xi_{2}^{n}$.