Extension of a Measure

Prove that the cardinality of the set of Lebesgue measurable subsets of $[0,1]$ is greater than that of the continuum.

Let $\mu$ be Lebesgue measure on $[0,1]$, and introduce an equivalence relation on the space of Lebesgue measurable subsets of $[0,1]$ by setting $A\sim B$ if $\mu \left(A\triangle B\right)=0$. Prove that the collection of equivalence classes has the cardinality of the continuum.

Let $\mu$ be a measure on $S$. Prove that the following conditions are equivalent if $S$ is a ring, and may not be equivalent if $S$ is a half-ring.

1. Countable additivity: $\mu \left(\coprod_{k=1}^{\infty }A_k\right)=\sum_{k=1}^{\infty }\mu \left(A_k\right)$.

2. Upper semicontinuity: if $A_1\supset A_2\supset A_3\supset \cdots$ and $A=\bigcap_{k=1}^{\infty }A_k$, then $\mu \left(A\right)=\lim \mu \left(A_k\right)$.

3. Lower semicontinuity: if $A_1\subset A_2\subset A_3\subset \cdots$ and $A=\bigcup_{k=1}^{\infty }A_k$, then $\mu \left(A\right)=\lim \mu \left(A_k\right)$.

4. Continuity: $\mu \left(\lim_{n\to \infty }A_k\right)=\lim_{n\to \infty }\mu \left(A_k\right)$.

Let $\mu$ be a countably additive measure on a half-ring $S\subset \mathcal P(X)$, and $\mu^*$ the corresponding outer measure on $\mathcal P(X)$.

Let $S$ be the half-ring of intervals of the form $[a,b)$ on $[0,1]$, and $\mathcal M$ the space constructed in Problem 95(a). Prove that $\mathcal M$ is connected and not compact.

Let $\left\{ E_n\right\}$ be a sequence of Lebesgue measurable subsets of the line. Are the limits supremum and infimum of the sequence $\left\{ E_n\right\}$ measurable sets (see Problem 82)?

Let $A_n$ be a sequence of measurable sets with $\sum \mu \left(A_n\right)<\infty$. Prove that $\mu \left(\limsup A_n\right)=0$.

Suppose that $X$ is the unit square in the plane and $S$ is the half-ring of rectangles in $X$ of the form

Define $m(T_{ab})=b-a$. Describe the explicit form of the Lebesgue extension of the measure $m$.

Under the conditions and with the notation of Problem 100 prove that the set $\widetilde T=\left\{ 0\leq x\leq 1,\ y=1/2\right\}$ is not measurable, and find its outer measure.

Suppose that a measure $\mu$ is defined on a half-ring $X$ with unit, and let $\mu^*$ be its outer measure. A set $A\subset X$ is said to be measurable in the Carathéodory sense if for any subset $Z\subset X$

Prove that $A$ is Lebesgue measurable if and only if it is measurable in the Carathéodory sense.

Let $\mu$ be a $\sigma$-additive measure defined on a half-ring. A set $A$ is called a set of $\sigma$-uniqueness for $\mu$ if:

1. there exists a $\sigma$-additive extension $\lambda$ of $\mu$ that is defined on $A$;

2. for every two such $\sigma$-additive extensions $\lambda_1$ and $\lambda_2$, $\lambda_1\left(A\right)=\lambda_2\left(A\right)$.

Suppose that each of the sets $X_n$, $n=1,2,3,\ldots$, consists of the digits $0,1,2,\ldots ,9$. Define a measure $\mu_n$ on $X_n$ by setting $\mu_n\left(Y\right)=(1/10)\operatorname {card}Y$. Let the measure $\mu$ on $X=\prod X_n$ be the product of the measures $\mu_n$. Prove that $\mu$ passes into the usual Lebesgue measure on $[0,1]$ under the mapping $\left\{ x_n\right\} \mapsto 0.x_1x_2x_3\ldots$ (infinite decimal fraction) of $X$ into $[0,1]$.