What is the difference between outer measure and Lebesgue measure?
We know that there are sets which are not Lebesgue measurable, whereas we know that outer measure is defined for any subset of $\mathbb{R}$.
lebesgue-measuremeasure-theoryreal-analysis
What is the difference between outer measure and Lebesgue measure?
We know that there are sets which are not Lebesgue measurable, whereas we know that outer measure is defined for any subset of $\mathbb{R}$.
Best Answer
The Lebesgue measure and Lebesgue outer measure coincide on Lebesgue measurable sets, which can be defined in several equivalent ways. Let $m$ and $m^*$ denote the Lebesgue measure and the Lebesgue outer measure respectively. These are some possible definitions of $A\subset\mathbb{R}^n$ being measurable:
The reason for the need of two different concepts is that neither of them is "perfect":