In Terence Tao's paper " Szemerdi's Regularity Lemma Revisited", the classic result is rephrased to Probabilistic terms instead of Graph Theoretical terms. In the partition step, the $\sigma$-algebra $\mathcal{B}_i$ is partitioned into sets.
It is not clear to me how this partition works. What is a partition of atoms of a $\sigma$-algebra? And if the sample space on which this algebra is defined is $A_1\times V_2$ for some $A_1\subseteq V_1$, how come the union of all this set is of size $|V_1|$?
Any clarification would help here.