Which set I say that it's measurable set, and which one is a Lebesgue measurable set !!! I read that the Set which is applied on Lebesgue measure is a sequence of small intervals. so what about the general measure. To be clear, what's the diff between the outer measure and Lebesgue outer measure?
2026-02-23 01:03:36.1771808616
Measurable and lebesgue measurable sets
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Given a $\sigma$-algebra $\mathcal{F}$ on a set $\Omega,$ we say that the sets $A\in\mathcal{F}$ are $\mathcal{F}$-measurable.
In this Wikipedia entry, you can see how an outer measure is defined, and how an outer measure gives rise to a $\sigma$-algebra of measurable sets, for which the measure equals the outer measure (by definition).
Both concepts can then be specialized to the case of Lebesgue outer measure/ Lebesgue measurable sets by taking the appropriate notion of outer measure.