An outer measure is a set function defined on some set X with the properties that empty returns 0, of sub additivity and of monotonicity. From this outer measure we get a sigma algebra of …

2015年12月14日 · An outer measure on X is a function $\theta : \mathcal{P}X \to [0,\infty]$ such that (i) $\theta(\emptyset)=0$ (ii) if $ A \subseteq B \subseteq X$ then $\theta(A) \leq \theta(B)$

2015年7月17日 · You ask whether outer measure is countably additive, and precisely because of this confusion I'm not sure whether you're asking about additivity for any disjoint sets or just for …

2021年7月29日 · Lebesgue outer measure - Trouble understanding proof that outer Lebesgue measure of an interval is its length 2 The meaning of $\mu^\ast(A)=\infty$, where $\mu^\ast$ is …

Some books define measurable to be when inner measure equals outer measure, which makes your question quite easy. :) $\endgroup$ – Dave L. Renfro Commented Jan 14, 2015 at 19:18

2015年12月16日 · This question comes from Bass, Ex 4.15. Given a finite measure space $(X, \mathcal{A}, \mu)$, we can define an outer measure $\mu^*$ as $$ \mu^*(A) := \inf\{ \mu(B) : …

Instead of trying to find some new measure which satisfies all 4 properties, he restricted to a smaller collection of sets (as in Julián's answer) called "measurable sets" for which outer …

Every measure $\mu$ is induced by an outer measure $\mu^{\ast}$ canonically obtained from $\mu$, but in general the $\sigma$-algebra of $\mu^{\ast}$-measurable sets can be strictly …

2018年2月19日 · $\begingroup$ I mean the outer measure that you get? @Ramen When pre-measures are used it is usually in the context of extensions: every pre-measure gives rise to …

You can arrange them to have any given positive outer measure you want. There are many threads on this site where Vitali sets were discussed, among which: Vitali-type set with given …