3  Measure Theory

3.1 Definitions

We denote the collection of subsets, or power set, of a set \(X\) by \(\Pcal(X).\)

The Cartesian product, or product, of sets \(X, Y\) is the collection of all ordered pairs

\[ X\times Y = \{(x,y): x\in X, y\in Y\}. \]

A topological space is a set equipped with a collection of open subsets that satisfies appropriate conditions.

Definition 3.1 (Topological Space) A topological space \((X, \Tcal)\) is a set \(X\) and a collection \(\Tcal \subset \Pcal(X)\) of subsets of \(X,\) called open sets, such that

  1. \(\emptyset, X \in \Tcal;\)

  2. If \(\{U_\alpha \in \Tcal: \alpha \in I \}\) is an arbitrary collection of open sets, then their union

    \[ \bigcup_{\alpha\in I} U_\alpha \in \Tcal \]

    is open;

  3. If \(\{U_i \in \Tcal: i=1,2,\dots,N \}\) is a finite collection of open sets,Then their intersection

    \[ \bigcap_{i=1}^N U_i \in \Tcal \]

    is open.

The complement of an open set in \(X\) is called a closed set, and \(\Tcal\) is called a topology on \(X.\)

A \(\sigma\)-algebra on a set \(X\) is a collection of subsets of a set \(X\) that contains \(\emptyset\) and \(X\), and is closed under complements, finite unions, countable unions, and countable intersections.

Definition 3.2 A \(\sigma\)-algebra on a set \(X\) is a collection \(\Acal\) of subsets of a set \(X\) such that:

  1. \(\emptyset, X \in \Acal;\)
  2. If \(A\in\Acal\) then \(A^c\in\Acal;\)
  3. If \(A_i\in\Acal\) then \[ \bigcup_{i=1}^\infty A_i \in \Acal, \quad \bigcap_{i=1}^\infty A_i \in \Acal. \]

Example 3.1 If \(X\) is a set, then \(\{\emptyset,X\}\) and \(\Pcal(X)\) are \(\sigma\)-algebras on \(X\); they are the smallest and largest \(\sigma\)-algebras on \(X\), respectively.

A measurable space \((X, \Acal)\) is an non-empty set \(X\) equipped with a \(\sigma\)-algebra \(\Acal\) on \(X.\)

Difference between a measurable space and \(\sigma\)-algebra:

  • The complement of a measurable set is measurable, but the complement of an open set is not, in general, open, excluding special cases such as the discrete topology \(\Tcal = \Pcal (X)\)

  • Countable intersections and unions of measurable sets are measurable, but only finite intersections of open sets are open while arbitrary (even uncountable) unions of open sets are open.

A measure \(\mu\) is a countably additive, non-negative, extended real-valued function defined on a \(\sigma\)-algebra.

A measure space \((X, \Acal, \mu)\) consist of a set \(X\), a \(\sigma\)-algebra \(\Acal\) on \(X\), and a measure \(\mu\) defined on \(\Acal.\) When \(\Acal\) and \(\mu\) are clear from the context, we will refer to the measure space \(X\).

An abstract probability space \((\Omega, \Fcal, \P)\)

  • \(\omega\in \Omega\) is called an outcome;

  • \(A\in \Fcal\) is called an event;

  • \(\P(A)\) is called the probability of \(A.\)

    \(\P(\Omega)=1\) the sum of probability of all possible outcomes is 1.

A random variable is any function \(X: \Omega \to \Xcal.\) We say that \(X\) has distribution \(P,\) and write \(X\sim P\), if

\[ \P(X\in B) = \P(\{\omega: X(\omega)\in B\}) = \P(B) \]

We say the real-valued random variable \(X\) is continuous if its distribution is absolutely continuous (with respect to the Lebesgue measure). If \(X\) is a random variable, then \(f(X)\) is also a random variable for any function \(f\).

The expectation of a random variable is defined as an integral with respect to \(\P\):

\[ \E[X] = \int X(\omega)\, \mathrm d \P(\omega), \]

and

\[ \E[f(X,Y)] = \int f(X(\omega), Y(\omega))\, \mathrm d \P(\omega). \]

A measure \(\mu\) on a measurable space \((X,\Acal)\) is a function

\[ \mu: \Acal \to [0, \infty] \]

such that

  1. \(\mu(\emptyset)=0;\)
  2. If \(\{A_i\in \Acal: i\in \N\}\) is a countable disjoint collection of sets in \(\Acal,\) then

\[ \mu \left( \bigcup_{i=1}^\infty A_i \right) = \sum_{i=1}^\infty \mu(A_i) \]

A measure \(\mu\) on a set \(X\) is

  • finite if \(\mu(X)<\infty,\) and
  • \(\sigma\)-finite if \(X=\bigcup_{n=1}^\infty A_n\) is a countable union of measurable sets \(A_n\) with finite measure, \(\mu(A_n)<\infty.\)

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