A question on the cardinality of a $\sigma$-field

Let $\gamma$ be a any class of subsets of $\Omega$ such that $\phi,\Omega\in\gamma$. We define $\gamma_0=\gamma$, and for any ordinal $\alpha>0$, inductively:

$$\gamma_\alpha = (\cup\set{\gamma_\beta:\beta<\alpha})’$$

Where $D’$ is the class of all countable unions of differences of sets in $D$.

What is $D’$? Assume $D = \set{\Omega, \phi}$ The difference $\Omega-\phi$ is the set $\Omega$. Assume $A\in{D}$ than $D’$ contain $\phi-A$, $\Omega-A=A^c$, $\phi-A=\phi$, etc.

But than - $A=\Omega-A^c\in{D’}$!

We than define $\frak{U}=\cup\set{\gamma_{\alpha}:\alpha<\beta}$

A proof for $\mu(\lim_{n\rightarrow\infty}{A})=\lim_{n\rightarrow\infty}{\mu(A_n)}$

Definition: If $\liminf A_n = \limsup A_n = A^{\star}$ we define $\lim_n A_n = A^{\star}$.

Problem: Assume $\mu$ a finite measure and $A_n\in\mathcal{F}, \forall{n}\in\mathbb{N}$. Show $\mu(\lim_{n\rightarrow\infty}{A})=\lim_{n\rightarrow\infty}{\mu(A_n)}$

An initial idea:

My initial idea was to use the following properties of $\liminf A_n$ and $\limsup A_n$:

1. $w\in\liminf{A_n} \iff \exist{n}$ such that $\forall{k\geq{n}}$ we have $w\in{A_k}$.

2. $w\in\limsup{A_n} \iff$ there are infinitely many $n$ such that $w\in{A_n}$.

However these propoerties do not reveal a connection between $\mu(\limsup{A_n})=\mu(\liminf{A_n})=\mu(A^{\star})$ and $\mu(A_n)$.

An explicit construction of a $\sigma$-field

$\sigma$-field definition and motivation:

Definition: A $\sigma$-field is a collection of subsets of a set $\Omega$ such that the following hold:

1. $\Omega \in F$.
2. if $A \in F$, so is $A^C \in F$.
3. For any countable collection of sets ${A_i}_{i=0}^\infty$ in $F$ their union is in $F$ as well.

The motivation for this discussion is based on the following problem by Ash:

“Let $A_1,…,A_n$ be arbitraty subsets of $\Omega$. Describe (explicitly) the smallest $\sigma$-field $\mathcal{F}$, containing $A_1,..,A_n$. How many sets are in there $\mathcal{F}$?”

The Mathematics of "Machi Koro"

“Machi Koro” is a strategy (and luck) board game where players try to gain assets and money.

Rules of the game

In each turn; each player tosses a die (or two, if permitted) and gains (or losses) money depending on the outcome of the die and his current assets.

Each player starts with the same set of assets:

The starting hand. Credit: Machi Koro - The Rules and Tactics

$\sigma$-fields and Measures

$\sigma$-fields:

The section begins with the definition of a sigma field (algebra) $F$:

Definition: A sigma field is a collection of subsets of a set $\Omega$ such that the following hold:

1. $\Omega \in F$.
2. if $A \in F$, so is $A^C \in F$.
3. For any countable collection of sets ${A_i}_{i=0}^\infty$ in $F$ their union is in $F$ as well.

Observations:

1. The empty set $\phi$ is also in F - since $\phi = \Omega^c \in F$.
2. Finite intersections are also in F - Using De Morgan, we may write - $(A^C \cup B^C)^C=A\cap B$
3. Using the same reasoning so is countable intersections.

Measures:

Definition: A measure $\mu$ is a non-negative countably additive set-function defined on a $\sigma$-field $F$ of a set $\Omega$:

Notes on Real Analysis and Probability (Ash)

Following the recommendations[1][2] of one Terry Tao, I’ve started this set of notes (or journal?) for my own enjoyment while reading through “Real Analysis and Probability” by Robert Ash.

I may jump to between chapters two and chapter five as the book is structured in such a way that one can do that.

Chapter 1:

Basic properties of sets:

0. If $A_1….A_k$ form an increasing sequence (in the sense $A_{1}\subset{A_{2}}…\subset{A_k}$), than their complements form a decreasing sequence - $A_{k}^c\subset{A_{i-2}^c}…\subset{A_1^c}$.

Proof: By induction. For $k=2$, $A_1\subset{}A_2$. Assume that $A_2^c\nsubseteq{}A_1^c$. We may conclude -