Outer measure: Difference between revisions

Revision as of 17:54, 20 October 2020

Definition. Let

X

be a nonempty set. An outer measure ^[1] on the set

X

is a function

\mu ^{*}:2^{X}\to [0,\infty ]

such that

$\mu ^{*}(\emptyset )=0$ ,
$\mu ^{*}(A)\leq \mu ^{*}(B)$ if $A\subseteq B$ ,
$\mu ^{*}(\cup _{j=1}^{\infty }A_{j})\leq \sum _{j=1}^{\infty }\mu ^{*}(A_{j}).$

Definition. A set

A\subset X

is

\mu ^{*}

-measurable if

\mu ^{*}(E)=\mu ^{*}(E\cap A)+\mu ^{*}(E\setminus A)

for all

E\subset X

.

↑ Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, second edition, Section 1.4

@@ Line 1: / Line 1: @@
-Let <math> X </math> be a nonempty set. An outer measure <ref name="Folland">Gerald B. Folland, ''Real Analysis: Modern Techniques and Their Applications, second edition'', Section 1.4 </ref> on the set <math> X </math> is a function <math> \mu^* : 2^X \to [0, \infty]</math> such that
+: '''Definition.''' Let <math> X </math> be a nonempty set. An outer measure <ref name="Folland">Gerald B. Folland, ''Real Analysis: Modern Techniques and Their Applications, second edition'', Section 1.4 </ref> on the set <math> X </math> is a function <math> \mu^* : 2^X \to [0, \infty]</math> such that
 * <math> \mu^* ( \emptyset) = 0 </math>,
 * <math> \mu^*(A) \leq \mu^*(B)</math> if <math> A \subseteq B</math>,
@@ Line 5: / Line 6: @@
-: '''Definition.'''
+: '''Definition.''' A set <math> A \subset X </math> is  <math> \mu^* </math>-measurable if <math>  \mu^*(E) = \mu^*(E \cap A) + \mu^* (E \setminus A)</math> for all  <math> E \subset X </math>.
-A set <math> A \subset X </math> is  <math> \mu^* </math>-measurable if <math>  \mu^*(E) = \mu^*(E \cap A) + \mu^* (E \setminus A)</math> for all  <math> E \subset X </math>.
 ==References==