Measures

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Definition

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} be a set and let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{M} \subseteq 2^X} be a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma} -algebra. Tbe structure $\left(X,{\mathcal {M}}\right)$ is called a measurable space and each set in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{M}} is called a measurable set. A measure on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (X, \mathcal{M})} (also referred to simply as a measure on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{M}} is understood) is a function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu: \mathcal{M} \rightarrow [0, \infty]} that satisfies the following criteria:

1. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu\left(\emptyset\right) = 0} ,
2. Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\{E_k\right\}_{k = 1}^{\infty}} be a disjoint sequence of sets such that each $E_{k}\in {\mathcal {M}}$. Then, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu\left(\cup_{k = 1}^{\infty} E_k\right) = \sum_{k = 1}^{\infty} \mu\left(E_k\right)} .

If the previous conditions are satisfied, the structure Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(X, \mathcal{M}, \mu\right)} is called a measure space.

Types of Measures

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(X, \mathcal{M}, \mu\right)} be a measure space.

• The measure $\mu$ is called finite if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu\left(X\right) < +\infty} .
• Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E \in \mathcal{M}} . If there exist $\left\{E_{k}\right\}_{k=1}^{\infty }\subseteq {\mathcal {M}}$ such that $E=\cup _{k=1}^{\infty }E_{k}$ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu\left(E_k\right) < + \infty} (for all $k\in \mathbb {N}$), then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} is $\sigma$-finite for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu} .
• If there exists $\left\{E_{k}\right\}_{k=1}^{\infty }\subseteq {\mathcal {M}}$ such that $X=\cup _{k=1}^{\infty }E_{k}$ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu\left(E_k\right) < + \infty} (for all $k\in \mathbb {N}$), then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu} is called Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma} -finite.

Properties

Let $\left(X,{\mathcal {M}},\mu \right)$ be a measure space.

1. Countable Additivity: Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\{E_k\right\}_{k = 1}^{n}} be a finite disjoint sequence of sets such that each $E_{k}\in {\mathcal {M}}$. Then, $\mu \left(\cup _{k=1}^{n}E_{k}\right)=\sum _{k=1}^{n}\mu \left(E_{k}\right)$. This follows directly from the defintion of measures by taking Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_{n+1} = E_{n+2} = ... = \emptyset} .
2. Monotonicity: Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E, F \in \mathcal{M}} such that $E\subseteq F$. Then, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu\left(E\right) \leq \mu\left(F\right)} .
3. Subadditivity: Let $\left\{E_{k}\right\}_{k=1}^{\infty }\subseteq {\mathcal {M}}$. Then, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu\left(\cup_{k = 1}^{\infty} E_k\right) \leq \sum_{k = 1}^{\infty} \mu\left(E_k\right)} .
4. Continuity from Below: Let $\left\{E_{k}\right\}_{k=1}^{\infty }\subseteq {\mathcal {M}}$ such that $E_{1}\subseteq E_{2}\subseteq ...$. Then, $\mu \left(\cup _{k=1}^{\infty }E_{k}\right)=\lim _{k\rightarrow \infty }\mu \left(E_{k}\right)$.
5. Continuity from Above: Let $\left\{E_{k}\right\}_{k=1}^{\infty }\subseteq {\mathcal {M}}$ such that $E_{1}\supseteq E_{2}\supseteq ...$ and $\mu \left(E'\right)<\infty$ for some $E'\in \left\{E_{k}\right\}_{k=1}^{\infty }$. Then, $\mu \left(\cap _{k=1}^{\infty }E_{k}\right)=\lim _{k\rightarrow \infty }\mu \left(E_{k}\right)$.

Examples

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References

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