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Diffstat (limited to 'pages/measure-and-integration/measure-theory')
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diff --git a/pages/measure-and-integration/measure-theory/borels-sets.md b/pages/measure-and-integration/measure-theory/borels-sets.md new file mode 100644 index 0000000..737a7c8 --- /dev/null +++ b/pages/measure-and-integration/measure-theory/borels-sets.md @@ -0,0 +1,33 @@ +--- +title: Borel Sets +parent: Measure Theory +grand_parent: Measure and Integration +nav_order: 2 +--- + +# {{ page.title }} + +{% definition Borel Sigma-Algebra, Borel Set %} +The *Borel σ-algebra* $\mathcal{B}(X)$ on a topological space $X$ is +the σ-algebra generated by its open sets. +The elements of $\mathcal{B}(X)$ are called *Borel(-measurable) sets*. +{% enddefinition %} + +That is, $\mathcal{B}(X) = \sigma(\mathcal{O})$, +where $\mathcal{O}$ is the collection of open sets in $X$. +It is also true that $\mathcal{B}(X) = \sigma(\mathcal{C})$, +where $\mathcal{C}$ is the collection of closed sets in $X$. + +{% definition Borel Function %} +If $(X,\mathcal{A})$ is a measure space +and $Y$ is a topological space, +then a function $f : X \to Y$ is called *measurable*, +or a *Borel function*, +if it is measurable with respect to $\mathcal{A}$ and +the Borel σ-algebra on $Y$. +{% enddefinition %} + +{% definition Borel Measure %} +A *Borel measure* on a topological space $X$ +is any measure on the Borel σ-algebra of $X$. +{% enddefinition %} diff --git a/pages/measure-and-integration/measure-theory/index.md b/pages/measure-and-integration/measure-theory/index.md new file mode 100644 index 0000000..575c945 --- /dev/null +++ b/pages/measure-and-integration/measure-theory/index.md @@ -0,0 +1,9 @@ +--- +title: Measure Theory +parent: Measure and Integration +nav_order: 1 +has_children: true +has_toc: false +--- + +# {{ page.title }} diff --git a/pages/measure-and-integration/measure-theory/measurable-maps.md b/pages/measure-and-integration/measure-theory/measurable-maps.md new file mode 100644 index 0000000..5b7a76e --- /dev/null +++ b/pages/measure-and-integration/measure-theory/measurable-maps.md @@ -0,0 +1,27 @@ +--- +title: Measurable Maps +parent: Measure Theory +grand_parent: Measure and Integration +nav_order: 3 +--- + +# {{ page.title }} + +{% definition Measurable Map %} +Suppose $(X,\mathcal{A})$ and $(Y,\mathcal{B})$ are measurable spaces. +We say that a map $f: X \to Y$ is *measurable* (with respect to $\mathcal{A}$ and $\mathcal{B}$) if +$f^{-1}(B) \in \mathcal{A}$ for all $B \in \mathcal{B}$. +{% enddefinition %} + +{% proposition %} +The composition of measurable maps is measurable. +{% endproposition %} + +It is sufficient to check measurability for a generator: + +{% proposition %} +Suppose that $(X,\mathcal{A})$ and $(Y,\mathcal{B})$ are measurable spaces, +and that $\mathcal{E}$ is a generator of $\mathcal{B}$. +Then a map $f : X \to Y$ is measurable iff +$f^{-1}(E) \in \mathcal{A}$ for every $E \in \mathcal{E}$. +{% endproposition %} diff --git a/pages/measure-and-integration/measure-theory/measures.md b/pages/measure-and-integration/measure-theory/measures.md new file mode 100644 index 0000000..637ab0c --- /dev/null +++ b/pages/measure-and-integration/measure-theory/measures.md @@ -0,0 +1,29 @@ +--- +title: Measures +parent: Measure Theory +grand_parent: Measure and Integration +nav_order: 4 +--- + +# {{ page.title }} + +{% definition %} +A *measure* on a σ-algebra $\mathcal{A}$ on a set $X$ +is mapping $\mu : \mathcal{A} \to [0,\infty]$ such that + +- $\mu(\varnothing) = 0$, +- for every sequence $(A_n)_{n \in \NN}$ of + pairwise disjoint sets $A_n \in \mathcal{A}$ + + $$ + \mu \bigg\lparen \bigcup_{n=1}^{\infty} A_n \! \bigg\rparen + = \sum_{n=0}^{\infty} \mu(A_n). + $$ +{% enddefinition %} + +{% definition Measure Space %} +A *measure space* is a triple $(X,\mathcal{A},\mu)$ of +a set $X$, +a σ-algebra $\mathcal{A}$ on $X$ +and a measure $\mu$ on $\mathcal{A}$. +{% enddefinition %} diff --git a/pages/measure-and-integration/measure-theory/sigma-algebras.md b/pages/measure-and-integration/measure-theory/sigma-algebras.md new file mode 100644 index 0000000..5d22f6b --- /dev/null +++ b/pages/measure-and-integration/measure-theory/sigma-algebras.md @@ -0,0 +1,50 @@ +--- +title: σ-Algebras +parent: Measure Theory +grand_parent: Measure and Integration +nav_order: 1 +--- + +# {{ page.title }} + +{% definition Sigma-Algebra, Measurable Space, Measurable Set %} +A *σ-algebra* on a set $X$ is a collection $\mathcal{A}$ of subsets of $X$ such that + +- $X$ belongs to $\mathcal{A}$, +- if $A \in \mathcal{A}$, then $X \setminus A \in \mathcal{A}$, +- the union of any countable subcollection of $\mathcal{A}$ belongs to $\mathcal{A}$. + +A *measurable space* is a pair $(X,\mathcal{A})$ consisting of +a set $X$ and a σ-algebra $\mathcal{A}$ on $X$. \ +The subsets of $X$ belonging to $\mathcal{A}$ are called *measurable sets*. +{% enddefinition %} + +{% example %} +On every set $X$ we have the σ-algebras $\braces{\varnothing,X}$ and $\mathcal{P}(X)$. +{% endexample %} + +{% proposition %} +If $\mathcal{A}$ is *σ-algebra* on a set $X$, then: + +- $\varnothing$ belongs to $\mathcal{A}$, +- if $A,B \in \mathcal{A}$, then $B \setminus A \in \mathcal{A}$, +- the intersection of any countable subcollection of $\mathcal{A}$ belongs to $\mathcal{A}$. +{% endproposition %} + +## Generated {{ page.title }} + +{% proposition Intersection of σ-Algebras %} +If $\braces{\mathcal{A}_i}$ is a family of σ-algebras on a set $X$, +then $\bigcap_i \mathcal{A}_i$ is a σ-algebra on $X$. +{% endproposition %} + +{% definition Generated σ-Algebras %} +Suppose $\mathcal{E}$ is any collection of subsets of a set $X$. +The *σ-algebra generated by $\mathcal{E}$*, denoted by $\sigma(\mathcal{E})$, is +defined to be the intersection of all σ-algebras on $X$ containing $\mathcal{A}$. +{% enddefinition %} + +By the previous proposition, $\sigma(\mathcal{E})$ is in fact a σ-algebra on $X$. + +## Products of {{ page.title }} + diff --git a/pages/measure-and-integration/measure-theory/signed-measures.md b/pages/measure-and-integration/measure-theory/signed-measures.md new file mode 100644 index 0000000..77b2416 --- /dev/null +++ b/pages/measure-and-integration/measure-theory/signed-measures.md @@ -0,0 +1,33 @@ +--- +title: Signed Measures +parent: Measure Theory +grand_parent: Measure and Integration +nav_order: 10 +--- + +# {{ page.title }} + +{% definition Signed Measure %} +A *signed measure* on a σ-algebra $\mathcal{A}$ on a set $X$ +is a mapping $\mu : \mathcal{A} \to [-\infty,\infty]$ such that +{: .mb-0 } + +- $\mu(\varnothing) = 0$, +- either there is no $A \in \mathcal{A}$ with $\mu(A) = -\infty$ + or there is no $A \in \mathcal{A}$ with $\mu(A) = \infty$, +- for every sequence $(A_n)_{n \in \NN}$ of + pairwise disjoint sets $A_n \in \mathcal{A}$ + {: .my-0 } + + $$ + \mu \bigg\lparen \bigcup_{n=1}^{\infty} A_n \! \bigg\rparen + = \sum_{n=0}^{\infty} \mu(A_n). + $$ +{% enddefinition %} + +{% definition Measure Space %} +A *measure space* is a triple $(X,\mathcal{A},\mu)$ of +a set $X$, +a σ-algebra $\mathcal{A}$ on $X$ +and a measure $\mu$ on $\mathcal{A}$. +{% enddefinition %} |