Lebesgue Measure of Intersection of two sets

Question

I have a one question relating to one property of Lebesgue Measures. If I have two sets, say $A \subset B $ and $B \subset C$ (closed or open) and Lebesgue measure is denoted by $\lambda$. Then my question is, what is the solution of following:

$$\lambda (C\backslash A \cap C\backslash B)=?$$

score 1 · Answer 1 · answered Nov 04 '13 at 13:51

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We have $(C\backslash A) \cap (C\backslash B)=C\backslash(A\cup B)=C\backslash B$. Then $$\lambda ((C\backslash A) \cap ( C\backslash B))=\lambda(C\backslash B)=\lambda(C)-\lambda(B)$$ if $C$ and $B$ are measurable.

answered Nov 04 '13 at 13:51

Mateus Sampaio

1,845

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thanks dear for this support – Lucky Nov 04 '13 at 14:40
This holds only if the measure of C is finite. – Sebastian Bechtel Jun 24 '15 at 19:46
You mean the measure of B has to be finite. You only get in troubles when subtracting infinity. So OP should add that assumption otherwise it's not true in general. – noctusraid Jan 18 '16 at 15:28

Lebesgue Measure of Intersection of two sets

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