Jordan Measure. Measurable Sets

Prove that in $R^{n}$ a cube of rank $k$ contains $\left(10^{l-k}-2\right)^{n}$ cubes of rank $l < k$ that do not intersect its boundary.

Prove that for a cube $Q$ of rank $k$ in $R^{n}$ there are:

Let $X \subset R^{n}$. Prove that:

Prove that an open cube of rank $k$ in $R^{n}$ is Jordan measurable and its measure equals the measure of the closed cube of rank $k$, i.e., $10^{-k n}$.

Using the definition of Jordan measure, prove measurability and find the measure of:

Using the definition of Jordan measure, prove the measurability of the sets:

Prove that the inner and outer Jordan measures of a bounded set are finite.

Prove that a Jordan measurable set is bounded.

Prove that:

Prove that the nonempty intersection of a closed $n$-dimensional cube with an open set in $R^{n}$ has positive inner measure.

Let $X$ be a closed set in $R^{n}$, and let $S_{k}(X)$ be its covering by cubes of rank $k$. Prove that every point of $X$ is an interior point of the set $S_{k}(X)$.

Let $X_{1} \subset X_{2} \subset R^{n}$. Prove that $\mu_{*}\left(X_{1}\right) \leqslant \mu_{*}\left(X_{2}\right)$ and $\mu^{*}\left(X_{1}\right) \leqslant \mu^{*}\left(X_{2}\right)$.

Let $X$ be a bounded set in $R^{n}$, $X_{1} \subset X$. Prove that:

Let $X$ be a bounded set in $R^{n}$, $X_{1} \subset X$. Prove that:

Give two disjoint sets $X_{1}$ and $X_{2}$ from $R^{n}$ such that

Let $X_{1}$ and $X_{2}$ be open sets in $R^{n}$. Prove that

Give two sets $X_{1}$ and $X_{2}$ from $R^{n}$ such that

Prove that if $\mu^{*}(X) = 0$, then $X$ is measurable and $\mu (X) = 0$.

Prove that the definition of Jordan measure given above, in the special case of a set of measure zero, is equivalent to the following: a set $X$ has Jordan measure zero if for any $\varepsilon > 0$ there exist a natural number $k$ and a finite covering $S_{k}(X)$ of the set $X$ by cubes of rank $k$ such that $\mu \left(S_{k}(X)\right) \leqslant \varepsilon$.

Prove that a finite set of points in $R^{n}$ has measure zero.

The sequence of points $x^{k} \in R^{n}, k \in N$, converges to a point in $R^{n}$. Prove that the set $\left\{ x^{k}: k \in N\right\}$ has measure zero.

Let $\alpha$ be a $k$-dimensional hyperplane in $R^{n}$ (a line when $k = 1$); let $X$ be a bounded subset of $\alpha$. Prove that $\mu (X) = 0$.

A set has measure zero. Using the definition of Jordan measure, prove that:

Let $X \subset R^{n}$, and suppose that for any $\varepsilon > 0$ there is a collection of $N$ Jordan measurable sets $X_{j}$ such that $X \subset \bigcup_{j = 1}^{N} X_{j}$ and $\sum_{j = 1}^{N} \mu \left(X_{j}\right) < \varepsilon$ ($N$ and $X_{j}$ in general depend on $\varepsilon$). Prove that $X$ is measurable and $\mu (X) = 0$.

Let $X \subset R^{n}, \partial X$ be the boundary of $X$, and let $\sigma_{k}(X)$ be the union of all cubes of rank $k$ contained in $S_{k}(X)$ but not belonging to $s_{k}(X), k = 0,1, \ldots$ Prove that $\partial X \subset \sigma_{k}(X) \subset S_{k}(\partial X)$.

Prove the criterion for measurability: for a set to be Jordan measurable, it is necessary and sufficient that it be bounded and that its boundary have Jordan measure zero.

Prove that the following are Jordan measurable:

Prove the monotonicity of the measure.

Prove the additivity of the measure.

Let $X_{1}$ and $X_{2}$ be Jordan measurable sets in $R^{n}$.

Prove that:

Prove that the closure $\bar{X}$ of a Jordan measurable set $X$ is Jordan measurable and $\mu (\bar{X}) = \mu (X)$.

Let $X$ be a measurable set, and let in $X$ be the set of all its interior points. Prove that in $X$ is measurable and $\mu ($ in $X) = \mu (X)$.

Let $X$ be a measurable set, and let in $X$ be the set of all its interior points. Prove that if $X_{1} \subset X \backslash$ in $X$, then $\mu \left(X_{1}\right) = 0$.

Let $X_{1} \subset R^{n}, \mu \left(X_{1}\right) = 0$. Prove that for any set $X \subset R^{n}$ the sets $X, X \cup X_{1}, X \backslash X_{1}$ are either all non-measurable or all measurable simultaneously, and in the latter case

Let the sets $X_{1}$ and $X_{2}$ be Jordan measurable in $R^{n}, X_{1} \subset \subset X_{2}$ and $\mu \left(X_{1}\right) = \mu \left(X_{2}\right)$. Prove that any set $X$ such that $X_{1} \subset X \subset X_{2}$ is Jordan measurable in $R^{n}$ and $\mu (X) = \mu \left(X_{1}\right) = \mu \left(X_{2}\right)$.

Let a Jordan measurable set $X \subset R^{n}$ be cut by an $(n-1)$-dimensional hyperplane $\alpha$ into two parts $X_{1}$ and $X_{2}$, i.e., all points of $X_{1}$ lie on one side of $\alpha$, all points of $X_{2}$ on the other, and $X = X_{1} \cup \alpha_{0} \cup X_{2}$, where $\alpha_{0} = \alpha \cap X, X_{1} \cap \alpha_{0} = X_{2} \cap \alpha_{0} = \varnothing$. Prove that $X_{1}$ and $X_{2}$ are Jordan measurable and $\mu \left(X_{1}\right)+\mu \left(X_{2}\right) = \mu (X)$.

Prove that:

A set $X \subset R^{n}$ is such that for any $\varepsilon > 0$ there exist two Jordan measurable sets $X^{\prime }$ and $X^{\prime \prime }$ such that $X^{\prime } \subset X \subset \subset X^{\prime \prime }$ and $\mu \left(X^{\prime \prime }\right)-\mu \left(X^{\prime }\right) < \varepsilon$. Prove that the set $X$ is Jordan measurable and

where $X^{\prime }$ and $X^{\prime \prime }$ range over all measurable sets contained in $X$ and containing $X$ respectively.

Prove the Jordan non-measurability of the set:

Give a Jordan non-measurable set in $R^{3}$.

Let $\left\{ X_{j}\right\}$ be a sequence of Jordan measurable sets in $R^{n}$ having no pairwise common interior points, and let $X = \bigcup_{j = 1}^{\infty } X_{j}$ be a bounded set. Prove that the series $\sum_{j = 1}^{\infty } \mu \left(X_{j}\right)$ converges and $\mu_{*}(X) = \sum_{j = 1}^{\infty } \mu \left(X_{j}\right)$.

Prove that for any open bounded nonempty set $X \subset R^{n}$ there is a sequence of cubes $Q_{j}$ of ranks $k_{j}, j \in N$, having no common interior points, such that $X = \bigcup_{j = 1}^{\infty } Q_{j}$ and

Let $X_{n}$ be Jordan measurable sets of measure zero, $n \in N$, and let $X = \bigcup_{n = 1}^{\infty } X_{n}$ be a Jordan measurable set.

Prove that $\mu (X) = 0$.

Give a countable collection of sets of Jordan measure zero whose union is not a set of Jordan measure zero.

Give a Jordan non-measurable set whose closure is Jordan measurable.

Prove that the union of two disjoint sets, one of which is measurable and the other Jordan non-measurable, is a set that is not Jordan measurable.

Give two non-measurable sets whose union is measurable.

Prove that every closed countable bounded set in $R^{n}$ is Jordan measurable and its measure equals zero.

Let $X^{\prime }$ be a Jordan measurable set in $R^{n}$. Prove that the cylinder $X = X^{\prime } \times [a ; b]$ is Jordan measurable in $R^{n+1}$ and $\mu_{n+1}(X) = (b-a) \cdot \mu_{n}\left(X^{\prime }\right)$.

Each of the projections of a set $X \subset R^{2}$ onto the coordinate axes is a measurable set in $R^{1}$. Must the set $X$ itself be measurable in $R^{2}$?

Give a bounded non-measurable set in $R^{2}$ such that its section by any line parallel to one of the coordinate axes is a measurable set in $R^{1}$.

Prove the measurability of every bounded set whose boundary is a union of a finite collection of sets, each of which is either the graph of a function continuous on a compact set, or part of a cylinder with a base of measure zero.

Prove the Jordan measurability of:

Let

Prove the measurability of the set $X \backslash \bigcup_{n = 1}^{\infty } X_{n}$ and find its measure.

Prove the measurability of the set

Prove that a rectifiable curve in $R^{n}$ has Jordan measure zero.

Let $\Omega$ be a closed bounded convex set in $R^{m}$, and let the functions $\varphi_{i}(y), i = 1, \ldots , n$, be continuously differentiable on $Q$. Prove that the $m$-dimensional surface given parametrically by $x_{i} = \varphi_{i}(y), i = 1, \ldots , n, y \in \Omega$, has Jordan measure zero in $R^{n}$.

Give a continuous curve $x = \varphi (t), y = \psi (t), a \leqslant t \leqslant b$, having positive measure in $R^{2}$.

Give a Jordan measurable set $X$ and a function $f$ continuous on it such that the set

is Jordan non-measurable.

Give a domain in $R^{2}$ that is Jordan non-measurable.

Let the function $f$ be continuous and nonnegative on the segment $[a ; b]$.

Prove that the curvilinear trapezoid

is Jordan measurable and

Let the function $r(\varphi )$ be continuous and nonnegative on the segment $[\alpha ; \beta ], 0 \leqslant \alpha < \beta < 2 \pi$. Prove that the sector

is Jordan measurable and

Let $X$ be a measurable set in $R^{n}, \alpha \in R^{n}$, and

be the set obtained by shifting $X$ by $a$. Prove that $X_{a}$ is measurable and $\mu \left(X_{a}\right) = \mu (X)$.

Let $X$ be a measurable set in $R^{n}, A-(n \times n)$-orthogonal matrix, $X_{A} = \left\{ x: x = A x^{\prime }, x^{\prime } \in X\right\}$ the set obtained by the orthogonal transformation $A$ of the set $X$ (a rotation, reflections, or their compositions). Prove that $X_{A}$ is measurable and $\mu \left(X_{A}\right) = \mu (X)$.

Prove that Jordan measure does not depend on the choice of rectangular Cartesian coordinate system.

Prove that for every measurable set there exist partitions of arbitrarily small mesh.

Prove that for every open measurable set there exist partitions of arbitrarily small mesh, all of whose elements have positive measure.

Let $X$ be an open measurable set, $\partial X$ its boundary, $X_{1} \subseteq \partial X$. Prove that for the set $X \cup X_{1}$ there exist partitions of arbitrarily small mesh, all of whose elements have positive measure.

Give a set of positive measure in $R^{n}$ for which there do not exist partitions of arbitrarily small mesh all of whose elements have positive measure.

Prove that for any measurable set $X$ there exists a sequence of nested partitions $\left\{ \tau_{n}(X)\right\} , \tau_{n+1}(X) \succ \succ \tau_{n}(X), n \in N$, with meshes tending to zero:

A set $X \subset R^{n}$ has Lebesgue measure zero if for any $\varepsilon > 0$ there is an at most countable collection of closed rectangular parallelepipeds $\left\{ P_{j}\right\}$ such that $X \subset \bigcup_{j = 1}^{\infty } P_{j}$ and $\sum_{j = 1}^{\infty } \mu \left(P_{j}\right) < \varepsilon$.

Prove that:

Prove that the set of all rational points of the segment $[0 ; 1]$, which is Jordan non-measurable in $R^{1}$ (see problem 40,1$)$), has Lebesgue measure zero in $R^{1}$.

Prove that the set:

Prove that the union of a countable collection of sets of Lebesgue measure zero also has Lebesgue measure zero.

Give a set $X$ of Lebesgue measure zero whose closure is not a set of Lebesgue measure zero (which, we note, is impossible for sets of Jordan measure zero).