Definition (Smallest $\sigma$-algebra)

For $\mathcal{M}\subseteq P(X)$ that is not necessarily a σ-algebra there is a smallest σ-algebra that contains $\mathcal{M}$, (aka the “σ-algebra generated by $\mathcal{M}$”): $$\sigma(\mathcal{M})=\bigcap_{\mathcal{A}\supseteq \mathcal{M}}\mathcal{A}$$where $\mathcal{A}$ are σ-algebras.