Title: Theory / Unimportant / Math / Measure theory - Fremlin{ / 25 Probability= radon

Question: Main difference between probability and the rest of measure theory ?
Answer: The codomain is more important than the domain. The domain can be changed without warning for the ease of the mathematician.
Page: 343
Chapter: 27
Section: 1

Name: A real-valued random variable on \(\Omega\)
Definition: A real-valued function defined on a conegligible subset of \(\Omega\) which is measurable with respect to the completion \(\hat\mu\) of \(\mu\).
Page: 344
Chapter: 27
Section: 1A
Variables: \((\omega,\Sigma,\mu)\) a probability space
Definition2: A member of \(\mathcal L^0(\mu)\).
Definition3: A real-valued function such that its restriction to a \(\Sigma\)-measurable conegligible set is measurable
Extra: Its measure does not have to be 1
Denoted by: \(X\)

Name: A probability space
Definition: A measure space \((\Omega,\Sigma,\mu)\) such that \(\mu(\Omega)=1\)
Page: 344
Chapter: 27
Section: 1A

Name: The mean of \(X\)
Definition: [$]\int X\dd\mu[/$]
Page: 344
Chapter: 27
Section: 1A
Variables: \((\omega,\Sigma,\mu)\) a probability space
\(X\) a real-valued random variable
Notation: \(\mathbb E(X)\)
Index: a
Name2: The expectation of \(X\)
Extra: Average?
Typ: Value in \([-\infty,+\infty]\)

Name: \(X\) has a finite expectation
Definition: \(X\) is integrable
Page: 344
Chapter: 27
Section: 1A
Variables: \((\omega,\Sigma,\mu)\) a probability space
\(X\) a real-valued random variable
Index: b

Name: The variance of \(X\)
Definition: \(\mathbb E(X-\mathbb E(X))^2\)
Page: 344
Chapter: 27
Section: 1A
Variables: \((\omega,\Sigma,\mu)\) a probability space
\(X\) a real-valued random variable
Index: c
Definition2: \(\mathbb E(X^2)-(\mathbb E(X))^2\)
Abbreviation: \(\text{Var}(X)\)

Definition: \(\hat\mu\{\omega\in \bigcap \text{dom }X_i\mid (X^1(\omega),\dots,X^n(\omega))\in S\}\)
Page: 344
Chapter: 27
Section: 1A
Variables: \((\omega,\Sigma,\mu)\) a probability space
\(X_1,\dots,X_n\) real-valued random variable,
\(S\subseteq\mathbb R^n\)
Notation: \(\text{Pr}((X^1,\dots,X^n)\in S)\)
Index: d
Assuming: \(S\) is Borel set

Definition: \(\hat\mu\{\omega:\omega\in\bigcap _{i\le n}\text{dom} X_i, \psi(X(\omega))\}\)
Page: 344
Chapter: 27
Section: 1A
Variables: \((\omega,\Sigma,\mu)\) a probability space
\(X_1,\dots,X_n\) real-valued random variables,
\(\psi(X_1,\dots,X_n)\) a predicate on \(X_i\)'s
Notation: \(\mathrm{Pr}(\psi(X_1,\dots,X_n))\)
Index: (d)
Assuming: \(\psi\) accepts a Borel measurable set