Let X = R (or any uncountable set). Define µ* : P(X) → R by { if A is countable µ*(A) = 1 if A is uncountable. Then show that µ* is an outer measure but not a measure.
Let X = R (or any uncountable set). Define µ* : P(X) → R by { if A is countable µ*(A) = 1 if A is uncountable. Then show that µ* is an outer measure but not a measure.
Let X = R (or any uncountable set). Define µ* : P(X) → R by { if A is countable µ*(A) = 1 if A is uncountable. Then show that µ* is an outer measure but not a measure.
Transcribed Image Text:Let X = R (or any uncountable set). Define µ* : P(X) → R by
{
if A is countable
µ*(A) =
1
if A is uncountable.
Then show that µ* is an outer measure but not a measure.
Branch of mathematical analysis that studies real numbers, sequences, and series of real numbers and real functions. The concepts of real analysis underpin calculus and its application to it. It also includes limits, convergence, continuity, and measure theory.
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