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How to prove that if the sets $A, B, C$ are all Lebesgue measurable, then

$$m(A \cup B \cup C)=m(A)+m(B)+m(C)-m(A \cap B)-m(A \cap C)-m(B \cap C)+m(A \cap B \cap C)$$

How to get an equivalent formula for $m(A)$, $m(B)$, and $m(C)$ so that I will no longer consider cases, such as when they have a common intersection, disjoint, or one of them is a subset of one or two of them? Also, if $m(A)$, $m(B) = \infty$, is $m(A\cap B)$ also $\infty$?

  • This is only for cases where $m(A\cup B \cup C)$ is finite. See https://math.stackexchange.com/q/2874519/442 – GEdgar Oct 02 '22 at 13:12
  • I think you have to assume some more stuff, the easiest being that $m(A),m(B),m(C)$ are all finite, because otherwise you could run into some undefined situtations. Also, to answer your question about whether $m(A)=m(B)=\infty$ implies that $m(A\cap B)=\infty$ or not, it is not true, but it can occur. Take for example $A=(-\infty,0]$, $B=[0,\infty)$. then $m(A)=m(B)=\infty$, but $m(A\cap B)=m({0})=0$. But also you could have, for example, that $A,B$ are such that $B\subseteq A$ and $m(A)=m(B)=\infty$. Then $m(A\cap B)=m(B)=\infty$. – Lorago Oct 02 '22 at 13:17
  • I am really trying to show or verify the formula for measure of the union of if n=3 sets, not the generalized formula –  Oct 02 '22 at 14:35
  • So I'm looking for hints on how to use the operations on sets to verify it –  Oct 02 '22 at 14:37

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