$\sigma$-algebra

Let $X$ be a set. A $\sigma$-algebra is a collection $M$ of subsets of $X$ such that

• $X\in M$

• If $A\in M$ then $X-A\in M$.

• If $A_1,A_2,A_3,\mathrm{\dots }$ is a countable subcollection of $M$, that is, $A_j\in M$ for $j=1,2,3,\mathrm{\dots }$ (the subcollection can be finite) then the union of all of them is also in $M$:

  $$\bigcup _{j=1}^{\mathrm{\infty }}{A}_i\in M.$$