# Motivation for and differences between properties of measure, outer measure

This is related to my measure theory class, but it's not homework. The motivation behind this post is to understand the big picture relationship between properties of measure and outer measure.

I understand the details of the how and what, but not the why. We have stated and proved many results in class, and the instructor explains nicely, but I need clarification.

So a measure $\mu$ is a set function on a field $\mathcal{F}$ over an underlying set $\Omega$ such that it satisfies:

$1.$ Nonnegativity, and can take on infinity as a value $\\$

$2.$ Zero measure of empty set $\\$

$3.$ Monotonicity $\\$

$4.$ Countable additivity for a disjoint subclass of $\mathcal{F}$, which implies finite additivity

$5.$ Countable subadditivity for not necessarily disjoint subclass of $\mathcal{F}.$

Questions: For $\#5,$ why is the equation having $\leq$ and not $<$? Under what condition(s) can equality hold? Also, my notes say that countable subadditivty implies finite subadditivity, but how?

An outer measure $\mu^{*}$ satisfies $\#1, \#2, \#3, \#5.$ According to my notes, monotonicity and C.S.A. are together weaker than countable additivity.

Questions: Why? Also, why is $\mu^{*}$ defined over $\mathcal{P}(\Omega)$ while $\mu$ is not? Also, why is $\mu^{*}\bigg|\mathcal{M}(\mu^{*})$ finitely additive? Given the finite additivity of $\mu^{*}$, why isn't it C.S.A?

Finally, what are the takeaways?

1. That countably additivity implies finite subadditivity is because after a finite amount of sets $n$ you can set the $n+1, n+2, \dots$ sets as the $\emptyset$ and this wont change the measure of the union nor the value of the sum.
2. On $\#5$ equality holds for all sets, then for a measure, and using some pairwise disjoint sets (by $4$) if we were to replace the $\leq$ by a strictly inequality then we would get
$$\sum_i \mu(A_i) = \mu(\cup A_i) < \sum_i \mu(A_i)$$
1. On the other hand $\mu^*$ is always defined on all subsets of $\mathcal{P}(\Omega)$, but fails to be countable additive on some disjoint family (non measurable sets), while $\mu$ is $\mu^*$ restricted to nicer sets (some sigma algebra), in which countably additivity always holds (when they're pairwise disjoint)
2. You already stated that an outer measure satisfies $\#5$, dont know why you're asking it again.