Finite Measurable Partitions and Their Refinements

In the settings of The Real Numbers: Standing Notation and Background and Measure Spaces and the Lebesgue Integral: Standing Notation, the measure space of the latter being instantiated at each use by the measure space named, let $(S,\mathcal{S})$ be a measurable space.

1. (Finite measurable partition)¶ A finite measurable partition of $S$ is a finite family $\mathcal{A}=(A_{1},\dots,A_{m})$, $m\in\mathbb{N}$, of sets $A_{i}\in\mathcal{S}$ with $A_{i}\cap A_{j}=\emptyset$ for all $i,j\in[m]$ with $i\neq j$, and whose union is $S$; the sets $A_{i}$ are its cells, and a cell may be empty.

2. (Refinement)¶ A finite measurable partition $\mathcal{B}=(B_{1},\dots,B_{l})$ of $S$ refines a finite measurable partition $\mathcal{A}=(A_{1},\dots,A_{m})$ of $S$ if for every $j\in[l]$ there is $i\in[m]$ with $B_{j}\subseteq A_{i}$.