Math Problems Vault

Problem 2.3.1

Let $p_{1}, \ldots , p_{n}>0$ . Fix $n$ real numbers $x_{1}<x_{2}<\cdots < x_{n}$ , and define

F(x)= \begin{cases} 0 & \text{ if } x<x_{1} \\ p_{1}+\cdots +p_{j} & \text{ if there exists } 1 \leqslant j \leqslant n \text{ such that } x_{j} \leqslant x<x_{j+1} \\ 1 & \text{ if } x \geqslant x_{n}\end{cases}

$\Delta_{F}$ is a measure on $\mathcal{A}_{1}$ , and

\Delta _{F}((a, b])= \begin{cases} \sum \left\{ p_{j}: j \text{ is such that } a<x_{j} \leqslant b\right\} & \text{ if }(a, b] \cap \left\{ x_{1}, \ldots , x_{n}\right\} \neq \varnothing \\ 0 & \text{ otherwise }\end{cases}

This is an important framework in discrete probability theory.

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Problem 2.3.2

This problem generalizes Example 4. Let $f: \mathbb {R}^{k} \rightarrow \mathbb {R}$ be such that $f$ is continuous and nonnegative. Further suppose that $\int_{-\infty }^{+\infty } \cdots \int_{-\infty }^{+\infty } f(\boldsymbol {t}) \mathrm{d} \boldsymbol {t}< +\infty$ . Define a function $F: \mathbb {R}^{k} \rightarrow \mathbb {R}$ by

F(\boldsymbol {x})=\int _{-\infty }^{x_{1}} \cdots \int _{-\infty }^{x_{k}} f(\boldsymbol {t}) \mathrm{d} \boldsymbol {t}, \quad \boldsymbol {x} \in \mathbb {R}^{k}

Then $\Delta_{F}$ is a measure on the semiring $\mathcal{A}_{k}$ , and for all $(\boldsymbol {a}, \boldsymbol {b}] \in \mathcal{A}_{k}$ we have

\Delta _{F}((\boldsymbol {a}, \boldsymbol {b}])=\int _{a_{1}}^{b_{1}} \cdots \int _{a_{k}}^{b_{k}} f\left(t_{1}, \ldots , t_{k}\right) \mathrm{d} t_{k} \cdots \mathrm{~ d} t_{1}

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Problem 2.3.3

(a)(b)

Let $F_{1}, \ldots , F_{k} \in \mathfrak {S}$ . For each $\boldsymbol {x} \in \mathbb {R}^{k}$ , write $F(\boldsymbol {x})= \prod_{i=1}^{k} F_{i}\left(x_{i}\right)$ .

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(a)

$F \in \mathfrak {S}_{k}$ , hence $\Delta_{F}$ is a measure on $\mathcal{A}_{k}$ .

(b)

$\Delta_{F}((\boldsymbol {a}, \boldsymbol {b}])=\prod_{i=1}^{k}\left[F_{i}\left(b_{i}\right)-F_{i}\left(a_{i}\right)\right]$ for all $(\boldsymbol {a}, \boldsymbol {b}] \in \mathscr {A}_{k}$ .

Problem 2.3.4

Suppose that $F_{i} \in \mathfrak {S}_{k_{i}}$ for $i=1, \ldots , n$ . Suppose that $F: \mathbb {R}^{\sum_{i=1}^{n} k_{i}} \rightarrow \mathbb {R}$ is such that

F\left(\boldsymbol {x}^{(1)}, \ldots , \boldsymbol {x}^{(n)}\right)=\prod _{i=1}^{n} F_{i}\left(\boldsymbol {x}^{(i)}\right)

for each $\boldsymbol {x}^{(1)} \in \mathbb {R}^{k_{1}}, \ldots$ , and $\boldsymbol {x}^{(n)} \in \mathbb {R}^{k_{n}}$ . Then $\Delta_{F}$ is a measure on $\mathcal{A}_{\sum_{i=1}^{n} k_{i}}$ . Also, $\Delta_{F}\left(A_{1} \times \cdots \times A_{n}\right)=\prod_{i=1}^{n} \Delta_{F_{i}}\left(A_{i}\right)$ for each $A_{1} \in \mathcal{A}_{1}, \ldots$ , and $A_{n} \in \mathcal{A}_{k_{n}}$ .

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Problem 2.3.5

For each $k \in \mathbb {N}$ we have the following: $\Delta_{F}(A)=0$ for all $A \in \mathcal{A}_{k}$ iff $F$ is a finite sum of functions, each function being a function of only at most $k-1$ of the coordinates.

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Remark.

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If $k=1$ , then a function of at most $k-1$ coordinates is a constant function.