Most results I have seen involves addition of measures

For example, let $m^*$ and $m$ be Lebesgue outer measure and Lebesgue measure respectively, then given $A = \bigcup\limits_{n = 1}^\infty E_n, E_i\cap E_j = \varnothing$

$m^*(A) \leq \sum\limits_{n = 1}^\infty E_n$

$m(A) = \sum\limits_{n = 1}^\infty E_n, \quad E_n \text{ measurable } \forall n$

Is there some nice results for "subtraction of sets" for Lebesgue measure and outer measure?

The only one I found was suppose you have some sort of decreasing sets $\{A_i\}, A_1 \supseteq A_2 \supseteq \ldots $, then given measure $m$, and $B_i = A_1\backslash A_i,$ and impose $m(A_1) < \infty$ $$ m(B_i) = m(A_1) - m(A_i)$$

Does above hold for outer measure as well? What about some other results?


1 Answer 1


For outer measure, if $A \subseteq B$ and $m^*(A)<\infty$, then $m^*(B\setminus A) \geq m^*(B)-m^*(A)$ because $A \cup (B\setminus A)=B$, so $m^*(B) \leq m^*(A)+m^*(B\setminus A)$ by the property of outer measure that you mentioned.


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