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Let $m^*$ be an outer measure on a set $X$. Is the following equality always true for every $A \subset X$ and measurable set of $ E\subset X$ ?

$m^*(E\cup A)+m^*(E\cap A) = m^*(E)+m^*(A)$

Note: I read Another thing on Outer measure, Caratheodory measure - proof . but in the book of Aliprantis that is always true . thanks

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nim
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1 Answers1

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$E$ being measurable gives, that for all $B$ $$ m^*(B) = m^*(B \cap E) + m^*(B \cap E^c)$$ applying this with $B = A \cup E$ gives $$ m^*(A \cup E) = m^*(E) + m^*(A \cap E^c) \iff m^*(A \cap E^c) = m^*(A \cup E) - m^*(E) \tag 1$$ and with $B = A$ $$ m^*(A) = m^*(A \cap E) + m^*(A \cap E^c)\iff m^*(A \cap E^c) = m^*(A) - m^*(A \cap E) \tag 2$$ Now equate (1) and (2).

martini
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