User:Aravind V R/sandbox1/Measure/Notes

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 with 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.

A Dirac measure is a measure δ_x on a set X (with any σ-algebra of subsets of X) defined for a given x ∈ X and any (measurable) set A ⊆ X by

${\displaystyle \delta _{x}(A)=1_{A}(x)={\begin{cases}0,&x\not \in A;\\1,&x\in A.\end{cases}}}$

where ${\displaystyle 1_{A}}$ is the indicator function of ${\displaystyle A}$. The Dirac measure is a probability measure

The counting measure ${\displaystyle \mu }$ on this measurable space ${\displaystyle (X,\Sigma )}$ is the positive measure ${\displaystyle \Sigma \rightarrow [0,+\infty ]}$ defined by

${\displaystyle \mu (A)={\begin{cases}\vert A\vert &{\text{if }}A{\text{ is finite}}\\+\infty &{\text{if }}A{\text{ is infinite}}\end{cases}}}$

for all ${\displaystyle A\in \Sigma }$, where ${\displaystyle \vert A\vert }$ denotes the cardinality of the set ${\displaystyle A}$. The counting measure on ${\displaystyle (X,\Sigma )}$ is σ-finite if and only if the space ${\displaystyle X}$ is countable.

Complete measure (or, more precisely, a complete measure space) is a measure space in which every subset of every null set is measurable (having measure zero). More formally, (X, Σ, μ) is complete if and only if

${\displaystyle S\subseteq N\in \Sigma {\mbox{ and }}\mu (N)=0\ \Rightarrow \ S\in \Sigma .}$

Let X be a locally compact Hausdorff space, and let ${\displaystyle {\mathfrak {B}}(X)}$ be the smallest σ-algebra that contains the open sets of X; this is known as the σ-algebra of Borel sets. A Borel measure is any measure μ defined on the σ-algebra of Borel sets. The real line ${\displaystyle \mathbb {R} }$ with its usual topology is a locally compact Hausdorff space, hence we can define a Borel measure on it. In this case, ${\displaystyle {\mathfrak {B}}(\mathbb {R} )}$ is the smallest σ-algebra that contains the open intervals of ${\displaystyle \mathbb {R} }$. While there are many Borel measures μ, the choice of Borel measure which assigns ${\displaystyle \mu ([a,b])=b-a}$ for every interval ${\displaystyle [a,b]}$ is sometimes called "the" Borel measure on ${\displaystyle \mathbb {R} }$.

Given a subset ${\displaystyle E\subseteq \mathbb {R} }$, with the length of an (open, closed, semi-open) interval ${\displaystyle I=[a,b]}$ given by ${\displaystyle l(I)=b-a}$, the Lebesgue outer measure ${\displaystyle \lambda ^{*}(E)}$ is defined as

${\displaystyle \lambda ^{*}(E)=\operatorname {inf} \left\{\sum _{k=1}^{\infty }l(I_{k}):{(I_{k})_{k\in \mathbb {N} }}{\text{ is a sequence of open intervals with }}E\subseteq \bigcup _{k=1}^{\infty }I_{k}\right\}}$.

The Lebesgue measure of E is given by its Lebesgue outer measure ${\displaystyle \lambda (E)=\lambda ^{*}(E)}$ if, for every ${\displaystyle A\subseteq \mathbb {R} }$,

${\displaystyle \lambda ^{*}(A)=\lambda ^{*}(A\cap E)+\lambda ^{*}(A\cap E^{c})}$.

The Borel measure agrees with the Lebesgue measure on those sets for which it is defined; however, there are many more Lebesgue-measurable sets than there are Borel measurable sets. The Borel measure is translation-invariant, but not complete.

Urysohn's lemma states that a topological space is normal if and only if any two disjoint closed subsets can be separated by a function.

Let (f_n) be a sequence of M-valued measurable functions, where M is a separable metric space, on some measure space (X,Σ,μ), and suppose there is a measurable subset A of finite μ-measure such that (f_n) converges μ-almost everywhere on A to a limit function f. The following result holds: for every ε > 0, there exists a measurable subset B of A such that μ(B) < ε, and (f_n) converges to f uniformly on the relative complement A \ B.