For a given stationary process $(X, \mathfrak{A}, m, T_{t})$, let $\mathfrak{B}$ be a sub-field of measurable sets and $\mathfrak{C}$ the sub-field of all invariable sets. The mutual independence of the two sub-fields $\mathfrak{B}$ and $\mathfrak{C}$ is a necessary and sufficient condition for the ergodicity of every random variable $\mathfrak{B}$-measurable and integrable.