To prove the statement, we need to show two things:

If the limit of the sequence of subsets En is equal to the set E, then the limit of the characteristic functions of En converges pointwise to the characteristic function of E.

If the limit of the characteristic functions of En converges pointwise to the characteristic function of E, then the limit of the sequence of subsets En is equal to the set E.

Here is the proof:

Suppose that lim En = E. Let χ_A(x) denote the characteristic function of a set A, that is, χ_A(x) = 1 if x ∈ A and χ_A(x) = 0 if x ∉ A. We need to show that lim χ_En(x) = χ_E(x) for all x ∈ X. Fix x ∈ X. If x ∈ E, then x belongs to En for infinitely many values of n. Therefore, χ_En(x) = 1 for infinitely many values of n, and hence lim χ_En(x) = 1 = χ_E(x). On the other hand, if x ∉ E, then x does not belong to En for all sufficiently large n. Therefore, χ_En(x) = 0 for all sufficiently large n, and hence lim χ_En(x) = 0 = χ_E(x). This shows that lim χ_En(x) = χ_E(x) for all x ∈ X, as required.

Now suppose that lim χ_En(x) = χ_E(x) for all x ∈ X. We need to show that lim En = E. First, note that if x ∈ E, then χ_E(x) = 1, and hence χ_En(x) = 1 for infinitely many values of n. Therefore, x ∈ En for infinitely many values of n, which implies that x ∈ lim En. On the other hand, if x ∉ E, then χ_E(x) = 0, and hence χ_En(x) = 0 for all sufficiently large n. Therefore, x ∉ En for all sufficiently large n, which implies that x ∉ lim En.

Thus, we have shown that lim En is a subset of E, and that every element of E is also an element of lim En. This means that lim En = E, as required.

Therefore, we have shown both directions of the “if and only if” statement, and hence the statement is proved.

Answer by ChatGPT: A Borel sigma algebra on the real line is the sigma algebra generated by all the open intervals on the real line, or equivalently, the smallest sigma algebra that contains all the open sets on the real line. The cardinality of the Borel sigma algebra on the real line is equal to the cardinality of the real numbers, which is known as the cardinality of the continuum and is denoted by $\mathfrak{c}$.

To see why this is the case, note that any open interval on the real line can be written as a countable union of disjoint open intervals with rational endpoints. Therefore, the Borel sigma algebra contains at least as many sets as there are real numbers, since each such interval corresponds to a unique pair of rational numbers. Conversely, any set in the Borel sigma algebra can be written as a countable union or intersection of open intervals, and so the cardinality of the Borel sigma algebra is at most the cardinality of the set of all countable sequences of open intervals on the real line, which has cardinality $\mathfrak{c}$.

Therefore, the cardinality of the Borel sigma algebra on the real line is $\mathfrak{c}$.

R: Since A and B are Jordan measurable sets, they are both bounded and have finite measure. Therefore, A∩B is bounded and has finite measure, making it Jordan measurable.

R: Proof: Let $A$ be a subset of real numbers and let $A^*$ be its closure. Since $A$ is a subset of $A^*$, it follows that $A^*$ is a superset of $A$. Therefore, the Jordan outer measure of $A^*$ is greater than or equal to the Jordan outer measure of $A$.

Now, let us consider the opposite direction. Since $A^*$ is the closure of $A$, it follows that $A^*$ contains all of the limit points of $A$. Therefore, any subset of $A^*$ which is contained in $A$ is also contained in $A^*$. This implies that the Jordan outer measure of $A^*$ is less than or equal to the Jordan outer measure of $A$.

Combining the two directions, we can conclude that the Jordan outer measure of $A$ and $A^*$ are equal.

Therefore, $A$ and its closure have the same Jordan outer measure.

Q.E.D.

First, let us define what a G delta set is: A subset of the real line is a G delta set if it can be written as a countable intersection of open sets.

Now, we need to show that E is Lebesgue measurable if and only if E is a G delta set with a null set removed.

Suppose $E$ is Lebesgue measurable. By definition, this means that for any $\epsilon > 0$, there exists an open set $O$ containing $E$ such that the Lebesgue measure of $O \setminus E$ is less than $\epsilon$.

Let $F$ be the intersection of all open sets containing $E$ and having measure less than $\epsilon$. Then $F$ is a $G_{\delta}$ set since it is the countable intersection of open sets. Also, $F$ contains $E$, and $F\setminus E$ has measure zero since $F$ is the intersection of all open sets containing $E$ and having measure less than $\epsilon$.

Conversely, suppose that $E$ is a $G_{\delta}$ set with a null set removed, i.e., $E = G\setminus N $ where $G$ is a $G_{\delta}$ set and $N$ is a null set. Then, since $G$ is a $G_{\delta}$ set, there exists a sequence of open sets $\{O_n\}$ such that $G = \cap O_n$. Let $ O_{n^{´}} = O_n \setminus N$, then $E = \cap O_{n^{´}}$ and $O_{n^{´}}$ is an open set since $N$ is a null set.

Since $N$ is a null set, its Lebesgue measure is zero. Therefore, for any $\epsilon > 0$, there exists an $N$ such that the Lebesgue measure of $N$ is less than $\epsilon /2$. Thus, for any $\epsilon > 0$, there exists an $O_{n^´}$ such that the Lebesgue measure of $O_{n^{'}}\setminus E$ is less than $\epsilon /2$. Then, for any $\epsilon > 0$, the Lebesgue measure of $O_{n^´} \Delta (E \cap O_{n^´})$ is less than $\epsilon/2$.

Now, let $O = \cap_n O_{n^´}. Then, $E \subset O$ since $E = \cap_n O_{n^´} and $O_{n^´}\subset O$ for all $n$. Also, for any $\epsilon > 0$, we have:

$$m(O\setminus E) = m(\cup_n (O_{n^´} \Delta (E \cap O_{n^´}))) \leq \sum_n m(O_{n^´} \Delta(E \cap O_{n^´})) \leq \epsilon.$$

where the last inequality follows from the fact that the series converges since $\cup_n (O_{n^´} \Delta (E \cap O_{n^´}))$ is a subset of $N$ which has measure less than $\epsilon /2$. Therefore, $E$ is Lebesgue measurable. Hence, we have shown that E is Lebesgue measurable if and only if E is a G delta set with a null set removed.