A Mild Glimpse Into Measure Theory (2)

by tzy9393

3. Measures and Outer measures

 

Let {(E,\mathscr{M})} be a measurable space.

Definition: A measure on {(E,\mathscr{M})} is a function {\mu:\mathscr{M}\rightarrow [0,\infty]} such that:
(1) {\mu(\emptyset)=0}
(2) For all families {\left\{A_{n}\right\}_{n\in\mathbb{N}}} of disjoint measurable sets,

\displaystyle \mu(\bigcup_{n\in\mathbb{N}} A_{n})=\sum_{n\in\mathbb{N}}\mu(A_{n})

Remarks:

  • {\mu(A)=+\infty} is allowed
  • Property (2) is called {\sigma}-additivity
  • We can also derive {\mu(\bigcup_{n=1}^{k} A_{n})=\sum_{n=1}^{k}\mu(A_{n})}

 

Proposition 6 : Let {A,B\in\mathscr{M}}, we have the following:
(1) If {A\subseteq B}, then {\mu(A)\leq\mu(B)}.
(2) If {\mu(A)<+\infty} and {A\subseteq B}, then {\mu(A\setminus B)=\mu(A)-\mu(B)}
(3) {\mu(A\cup B)=\mu(A)+\mu(B)-\mu(A\cap B)}
(4) If {A_{n}\in\mathscr{M}} for all {n\in\mathbb{N}} and {A_{n}\subseteq A_{n+1}} for all {n\in\mathbb{N}}, then {\mu(\bigcup_{n\in\mathbb{N}}A_{n})=\lim_{n\rightarrow\infty}\mu(A_{n})}.
(5) If {B_{n}\in\mathscr{M}} for all {n\in\mathbb{N}} and {B_{n}\supseteq B_{n+1}} for all {n\in\mathbb{N}}, then {\mu(\bigcap_{n\in\mathbb{N}}B_{n})=\lim_{n\rightarrow\infty}\mu(B_{n})}.
(6) If {A_{n}\in\mathscr{M}}, then {\mu(\bigcup_{n\in\mathbb{N}}A_{n})\leq\sum_{n\in\mathbb{N}}A_{n}}.

To be updated.

 

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