Chapter 1, Basic Measure Theory Video Solutions, Probability Theory: A Comprehensive Course

Problem 1

Let $\mathcal{A}$ be a semiring. Show that any countable (respectively finite) union of sets in $\mathcal{A}$ can be written as a countable (respectively finite) disjoint union of sets in $\mathcal{A}$.

Angelo Rendina

Numerade Educator

06:04

Problem 2

Give a counterexample that shows that, in general, the union $\mathcal{A} \cup \mathcal{A}^{\prime}$ of two $\sigma$-algebras need not be a $\sigma$-algebra.

Mengchun Cai

Numerade Educator

01:19

Problem 3

Let $\left(\Omega_{1}, d_{1}\right)$ and $\left(\Omega_{2}, d_{2}\right)$ be metric spaces and let $f: \Omega_{1} \rightarrow \Omega_{2}$ be an arbitrary map. Denote by $U_{f}=\left\{x \in \Omega_{1}: f\right.$ is discontinuous at $\left.x\right\}$ the set of points of discontinuity of $f$. Show that $U_{f} \in \mathcal{B}\left(\Omega_{1}\right)$.
Hint: First show that for any $\varepsilon>0$ and $\delta>0$ the set
$$
U_{f}^{\delta, \varepsilon}:=\left\{x \in \Omega_{1}: \text { there are } y, z \in B_{\varepsilon}(x) \text { with } d_{2}(f(y), f(z))>\delta\right\}
$$
is open (where $\left.B_{\varepsilon}(x)=\left\{y \in \Omega_{1}: d_{1}(x, y)<\varepsilon\right\}\right) .$ Then construct $U_{f}$ from such $U_{f}^{\delta, \varepsilon}$

Linh Vu

Numerade Educator

09:01

Problem 4

Let $\Omega$ be an uncountably infinite set and $\mathcal{A}=\sigma(\{\omega\}: \omega \in \Omega)$. Show that
$$
\mathcal{A}=\left\{A \subset \Omega: A \text { is countable or } A^{c} \text { is countable }\right\}
$$

Mengchun Cai

Numerade Educator

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Problem 5

Let $\mathcal{A}$ be a ring on the set $\Omega$. Show that $\mathcal{A}$ is an Abelian algebraic ring with multiplication " $\cap$ " and addition " $\triangle "$.

Nick Johnson

Numerade Educator