Example

If $\mu$ is a measure on a locally compact space $X$ then the linear combinations, with real coefficients, of the characteristic functions of integrable sets form an algebra $\mathscr{A}$, because, for any two integrable sets $M, N$, the function $\varphi_M \varphi_N = \varphi_{M \cap N}$ is integrable (No. 5, Prop. 7); it then follows from Defs. 2 and 3 that the set of integrable subsets of $X$ is a clan.