If A and B are disjoint subsets of real numbers, and one of them is measurable can we say m*(A U B)=m*(A)+m*(B)? I am unable to find counter example. I feel this is not true.
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1$\begingroup$ Does $m^*$ here denote (Lebesgue) outer measure? $\endgroup$– Robin ChapmanCommented Jul 7, 2010 at 19:03
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$\begingroup$ yes m* is Lebesgue outer measure. $\endgroup$– khimji riyanCommented Jul 7, 2010 at 19:25
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$\begingroup$ The title of this question should be made more specific. $\endgroup$– j.c.Commented Jul 8, 2010 at 0:51
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$\begingroup$ I am voting to close as "off topic", because of the level. (I know from your comment at your other question that when you asked this you weren't clear on the purpose of MO, and I won't repeat what was said there.) $\endgroup$– Jonas MeyerCommented Jul 8, 2010 at 3:07
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1 Answer
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This is true. If for example A is measurable it is measurable in the sense of Caratheodory so that
For every set C we will have
$m*(C) = m*(C\cap A) + m*(C \setminus A)$.
This with $C=A \cup B$ is your assertion
