User talk:Mindey/MathNew

Here will be newly known math formulae.

Definition of Conditional Expectation

The conditional expectation of the r.v. $X$ given $g$ (a subset of sigma field on which $X$ is defined) is a r.v. $Y$ satisfying:

(i) $Y\in g$ ($Y$ is a r.v. whose value can be determined from the information in $g$)

(ii) $\mathbb {E} (X\mathbf {1} _{A})=\mathbb {E} (Y\mathbf {1} _{A})$ for every $A\in g$.

Any r.v. $Y$ satisfying (i)-(ii) is denoted as $\mathbb {E} (X|g)$.

(Observation: actually, Y is not a r.v., but the sigma field. What is meant, is that "$Y$" r.v. defines the expectation, by its pairs of "element in sigma field, and probability assigned" (which allows then to find the mean, and it can be any pairs that produce that expectation.))

Radon-Nikodyn theorem

Definition of absolute continuity of a measure. A measure $\mu$ on measurable space $(\mathbb {X} ,\chi )$is absolutely continuous with respect to a measure $\nu$ (we write $\mu \ll \nu$) $\Leftrightarrow \mu (A)=0\Rightarrow \nu (A)=0$ (This is true if and only if there is a non-negative function $f:\mathbb {X} \rightarrow \mathbb {R} ^{+}\cup \{0\}$, such that $\nu (A)=\int _{A}f(x)d\mu (x)$.)

Theorem: If $\mu \ll \nu$, then there exists a measurable function $f:\mathbb {X} \mapsto \mathbb {R} _{+}\cup \{0\}$ such that $\nu (A)=\int _{A}f(x)d\mu (x)$.

σ-finite measure

[Wikipedia] A positive (or signed) measure μ defined on a σ-algebra Σ of subsets of a set X is called finite if μ(X) is a finite real number (rather than ∞). The measure μ is called σ-finite if X is the countable union of measurable sets of finite measure. A set in a measure space is said to have σ-finite measure if it is a countable union of sets with finite measure. — Preceding unsigned comment added by 128.211.173.130 (talk) 00:14, 23 August 2012 (UTC)

Co-countable set

[Wikipedia] Cocountable subset of a set X is a subset Y whose complement in X is a countable set. — Preceding unsigned comment added by 128.211.173.130 (talk) 00:36, 23 August 2012 (UTC)

Gδ set

In the mathematical field of topology, a G_δ set is a subset of a topological space that is a countable intersection of open sets.

Basic properties

• The complement of a G_δ set is an F_σ set.

Fσ set

In mathematics, an F_σ set (said F-sigma set) is a countable union of closed sets. — Preceding unsigned comment added by 128.211.178.4 (talk) 14:54, 29 August 2012 (UTC)

"Open" is defined relative to a particular topology

As a concrete example of this, if U is defined as the set of rational numbers in the interval (0, 1), then U is an open subset of the rational numbers, but not of the real numbers. This is because when the surrounding space is the rational numbers, for every point x in U, there exists a positive number a such that all rational points within distance a of x are also in U. On the other hand, when the surrounding space is the reals, then for every point x in U there is no positive a such that all real points within distance a of x are in U (since U contains no non-rational numbers).