Sets of Jordan measure zero

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 $X$ 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 $X_{n}$ 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.

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

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 $R^{2}$.

Let $E$ be a set of Jordan content 0 and $A \subset E$. Prove that $A$ is Jordan measurable and $c(A)=0$.

Let $A$ be a line segment in the $x y$-plane. Prove that $A$ has content 0 .

Let $f:[a, b] \rightarrow \mathbb {R}$ be a continuous function. Show that the graph $\left\{ (x, f(x)): x \in [a, b]\right\}$ is Jordan measurable in $\mathbb {R}^{2}$ with Jordan content zero.

Prove or disprove: a Jordan set in $\mathbb {R}^{2}$ has content 0 if and only if it is nowhere dense.

Let $f$ be a bounded function on a Jordan set $D$, let $A \subset D$ be the set of all the points on which $f$ fails to be continuous, and suppose that $A$ has content 0 . Prove that $f$ is integrable on $D$.

Let $X$ be a proper subspace of $\mathbb {R}^{n}$. Prove that $m\left(X\right)=0$, where $m$ is the $n$-dimensional Lebesgue measure.

Let $f: \mathbb {R} \rightarrow \mathbb {R}$ be continuous. Determine the measure of its graph.

Let $x \in \mathbb {R}$ and $k \geqslant 2$. Then $\lambda (\left\{ x\right\} )=\lambda_{k}\left(\left\{ x\right\} \times \mathbb {R}^{k-1}\right)=0$. Next, for any $j \in \left\{ 1, \ldots , k-1\right\}$ and $\boldsymbol {x} \in \mathbb {R}^{j}$ we have $\lambda_{k}\left(\left\{ \boldsymbol {x}\right\} \times \mathbb {R}^{k-j}\right)=0$.

Let $k \geqslant 2$. There exists an uncountable set $U \in \mathscr {B}^{k}$ with $\lambda_{k}(U)=0$.

Let $k \geqslant 2$. Every line is in $\mathscr {B}^{k}$, has $\lambda_{k}$-measure zero, and hence has $\bar{\lambda }_{k}$-measure.