# Measurable and lebesgue measurable sets

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?

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).