TITLE: Is a measure for a sigma algebra determined by its values for a generator of the sigma algebra?
QUESTION [18 upvotes]: If the value of a measure on any subset in a generator of a sigma algebra is known, will the measure for the sigma algebra also be uniquely determined? Thanks!

REPLY [1 votes]: Caratheodory’s measure extension theorem.:
Theorem 1.41 (Caratheodory). Let $A ⊂ 2^Ω$ be a ring and let $µ$ be a $σ$-finite
premeasure on $A$. There exists a unique measure $µ'$ on $σ(A)$ such that $µ'(E) = µ(E)$
for all $E ∈ A$. Furthermore, $µ'$ is $σ$-finite.
p. 19 of http://www.math.unipd.it/~daipra/didattica/galileiana-15/Parte-I.pdf
(The same is given on p. 14 of here: http://www.mathematik.uni-kl.de/~wwwstoch-alt/skripte/basicmeasuretheory_skript.pdf and here: https://en.wikipedia.org/wiki/Carath%C3%A9odory%27s_extension_theorem )
premeasure is a $σ$-additive set function $\mu$ to $[0,\infty]$ with $\mu(\emptyset)=0$.
Ring contains $\emptyset$ and is closed w.r.t. unions of two elements and complements.
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