Definition and Construction of a Measure and its Basic Properties

If $\Omega$ is countable and $\mu$ is defined on $\mathcal{P}(\Omega )$ by: $\mu (A) =$ number of points of $A$, show that $\mu$ is a measure. Furthermore, $\mu$ is finite or $\sigma$-finite, depending on whether $\Omega$ is finite or denumerable, respectively. (This measure is called the counting measure.)

Refer to the field $\mathcal{C}$ of Example 4 in Chapter 1 and on $\mathcal{C}$, define the set function $P$ as follows: $\mathbb {P}\left(A\right) = 0$ if $A$ is finite, and $\mathbb {P}\left(A\right) = 1$ if $A^{c}$ is finite. Then show that

Refer to the $\sigma$-field $\mathcal{C}$ of Example 8 in Chapter 1 and on $\mathcal{C}$, define the function $P$ as follows: $\mathbb {P}\left(A\right) = 0$ if $A$ is countable, and $\mathbb {P}\left(A\right) = 1$ if $A^{c}$ is countable. Then show that $P$ is a probability measure.

Let $A_{n}, n = 1,2, \ldots$ be events in the probability space $(\Omega , \mathcal{A}, P)$ such that $\mathbb {P}\left(A_{n}\right) = 1$ for all $n$. Then show that $\mathbb {P}\left(\cap_{n = 1}^{\infty } A_{n}\right) = 1$.

Let $\left\{ A_{i}, i \in I\right\}$ be an uncountable collection of pairwise disjoint events in the probability space $(\Omega , \mathcal{A}, P)$. Then show that $\mathbb {P}\left(A_{i}\right) > 0$ for countably many $A_{i} \mathrm{~ s}$ only.

Let $\Omega$ be an infinite set (countable or not) and let $\mathcal{A}$ be the discrete $\sigma$-field. Let $\left\{ \omega_{1}, \omega_{2}, \ldots \right\} \subset \Omega$, and with each $\omega_{n}$, associate a nonnegative number $p_{n}$ (such that $\sum_{n = 1}^{\infty } p_{n} \leq \infty$). On $\mathcal{A}$, define the set function $\mu$ by: $\mu (A) = \sum_{\omega_{n} \in A} p_{n}$. Then show that $\mu$ is a measure on $\mathcal{A}$.

In the measure space $(\Omega , \mathcal{A}, \mu )$ a set $A \in \mathcal{A}$ is called an atom, if $\mu (A) > 0$ and for any $B \subseteq A$ with $B \in \mathcal{A}$, it follows that $\mu (B) = 0$, or $\mu (B) = \mu (A)$. In reference to Exercise 6 , identify the atoms of $\mathcal{A}$.

In any measure space $(\Omega , \mathcal{A}, \mu )$ and with any $A_{n} \in \mathcal{A}, n = 1,2, \ldots$, show that

provided $\mu \left(\bigcup_{j = n}^{\infty } A_{j}\right) < \infty$ for some $n$. In Exercises 9 and 10, show that the set function $\mu^{\circ }$ defined on $\mathcal{P}(\Omega )$ is, indeed, an outer measure.

$\Omega$ is an arbitrary set, $\omega_{0}$ is a fixed point of $\Omega$, and $\mu^{\circ }$ is defined by: $\mu^{\circ }(A) = I_{A}\left(\omega_{0}\right)$.

$\Omega$ is the set of 100 points arranged in a square array of 10 columns, each with 10 points, and $\mu^{\circ }$ is defined by: $\mu^{\circ }(A) =$ number of columns that contain at least one point of $A$.

For an arbitrary set $\Omega$ containing at least two points, consider the trivial field $\mathcal{F} = \left\{ \varnothing , \Omega \right\}$, and let (the finite measure) $\mu$ be defined on $\mathcal{F}$ by: $\mu (\varnothing ) = 0$ and $\mu (\Omega ) = 1$.

Let $\Omega = \left\{ \omega_{1}, \omega_{2}, \omega_{3}, \omega_{4}\right\}$, let $\mathcal{C} = \left\{ \oslash , \left\{ \omega_{1}, \omega_{2}\right\} , \left\{ \omega_{1}, \omega_{3}\right\} , \left\{ \omega_{2}, \omega_{4}\right\} , \left\{ \omega_{3}, \omega_{4}\right\} , \Omega \right\}$, and define $\mu$ on $\mathcal{C}$ as follows:

Next, on $\mathcal{P}(\Omega )$, define the measures $\mu_{1}$ and $\mu_{2}$ by taking

Then show that

Let $\Omega = \left\{ 0,1,2, \ldots \right\}$, let $A = \left\{ 1,3,5, \ldots \right\}$, and let $\mathcal{C} = \left\{ \varnothing , A, A^{c}, \Omega \right\}$. Let $\mu$ be the counting measure on $\mathcal{C}$, and let $\mu_{1}, \mu_{2}$ be defined on $\mathcal{P}(\Omega )$ by

Then show that

Construct additional examples to illustrate the points made in Exercises 11(ii), 12 , and 13.

Consider the measure space $(\Omega , \mathcal{A}, \mu )$, and define the classes $\mathcal{A}^{*}$ and $\overline{\mathcal{A}}$ as follows:

Then show that

Refer to the classes $\mathcal{A}^{*}$ and $\overline{\mathcal{A}}$, defined in Exercise 15, and show that $\overline{\mathcal{A}}$ (and hence $\mathcal{A}^{*}$) is a $\sigma$-field.

Refer to Exercise 15 and on $\mathcal{A}^{*}$, define $\mu^{*}$ by: $\mu^{*}(A \Delta M) = \mu (A)$.

If $(\Omega , \mathcal{A}, \mu )$ is a measure space, by means of an example, show that there may be subsets $M \subset N$, where $N$ are null sets in $\mathcal{A}$, with $M \notin \mathcal{A}$. (If it so happens, however, that for every $M \subseteq N$ for every null $N \in \mathcal{A}$, then $M \in \mathcal{A}$, we say that $\mathcal{A}$ is complete with respect to $\mu$.)

If $\mu^{\circ }$ is an outer measure and $\mathcal{A}^{\circ }$ is the $\sigma$-field of $\mu^{\circ }$-measurable sets, then show that $\mathcal{A}^{\circ }$ is complete (with respect to $\mu^{\circ }$).

Consider the measure space $(\mathbb {R}, \mathcal{B}, \mu )$ where $\mu$ (finite interval) $< \infty$, and for $c \in \mathbb {R}$, let $F_{c}: \mathbb {R} \rightarrow \mathbb {R}$ be defined by

Then, by means of an example, show that $F_{c}$ need not be left-continuous.

Consider the measurable space $(\Omega , \mathcal{A})$, and let $\mu$ be a set function defined on $\mathcal{A}$ and taking nonnegative values, and such that $\mu (\oslash ) = c$ with $0 < c < \infty$. Then show that $\mu$ cannot be additive.

Let $\mu_{1}, \ldots , \mu_{n}$ be $\sigma$-finite measures on $(\Omega , \mathcal{A})$ and set $\mu = \mu_{1}+\cdots +\mu_{n}$ (in the sense that $\mu (A) = \sum_{i = 1}^{n} \mu_{i}(A), A \in \mathcal{A}$). Then show that $\mu$ is also $\sigma$-finite.

In the probability space $(\Omega , \mathcal{A}, P)$, show that

Consider the probability space $(\Omega , \mathcal{A}, P)$, where $\Omega = \left\{ \omega_{1}, \omega_{2}, \omega_{3}, \omega_{4}\right\} , \mathcal{A}$ is the discrete $\sigma$-field, and $P$ is defined by

For $n = 1,2, \ldots$, define $A_{n}$ by: $A_{2 n-1} = \left\{ \omega_{1}, \omega_{2}\right\} , A_{2 n} = \left\{ \omega_{2}, \omega_{3}\right\}$. Determine the: $\underline{\lim }_{n \rightarrow \infty } A_{n}, \overline{\lim }_{n \rightarrow \infty } A_{n}$ and compute the probabilities:

Let $\Omega = \left\{ \omega_{1}, \omega_{2}, \ldots \right\}$ and let $\mathcal{A}$ be a $\sigma$-field of subsets of $\Omega$. Then show that

Suppose that $\Omega = \left\{ \omega_{1}, \omega_{2}, \ldots \right\}$, and let $\mathcal{A}$ be the discrete $\sigma$-field. On $\mathcal{A}$, define the set function $\mu$ by

Then show that

Let $\Omega = \left\{ 1,2, \ldots \right\}$, and for any $A \subseteq \Omega$, let $a = \sup A$. On the discrete $\sigma$-field, define $\mu^{0}$ by

Then

Recall that $m$ is a median of a r.v. $X$ if $\mathbb {P}\left(X \leq m\right) \geq \frac{1}{2}$ and $\mathbb {P}\left(X \geq m\right) \geq \frac{1}{2}$. Then show that, if $m$ is a median of the r.v. $X$, then $-m$ is a median of the r.v. $-X$.

Recall that a r.v. $X$ is said to be symmetric about 0 , if $X$ and $-X$ have the same distribution. If $X$ is symmetric (about 0), then show that 0 is a median of $X$; i.e., $\mathbb {P}\left(X \leq 0\right) \geq \frac{1}{2}$ and $\mathbb {P}\left(X \geq 0\right) \geq \frac{1}{2}$.

Let $\mu^{0}$ be an outer measure, and suppose that $\mu^{0}(A) = 0$ for some $A \subset \Omega$. Then show that $\mu^{0}(A \cup B) = \mu^{0}(B)$ for every $B \subseteq \Omega$.

Consider the measure space $(\Omega , \mathcal{A}, \mu )$, and suppose that $\mathcal{A}$ is complete with respect to $\mu$. Let $f, g: \Omega \rightarrow \mathbb {R}$ be such that $\mu (f \neq g) = 0$. Then show that, if one of $f$ or $g$ is measurable, then so is the other.

Consider the measure space $(\Omega , \mathcal{A}, \mu )$, and let $\mathcal{A}$ be complete with respect to $\mu$. Let $A \in \mathcal{A}$ with $\mu (A) = 0$, and let $f$ be an arbitrary function $f: A \rightarrow \mathbb {R}$. Then $f$ is measurable.

Refer to Example 2 and: