  This expository article is based on two lectures given by the first author at
the Fields Institute in the Fall 2021 Thematic Program on Trends in Pure and
Applied Model Theory.
  We give a detailed proof of a qualitative version of the Mallaris-Shelah
regularity lemma for stable graphs using only basic local stability theory and
an ultraproduct construction. This proof strategy was first established by
Malliaris and Pillay, and later simplified by Pillay. We provide some further
simplifications, and also explain how the pseudofinite approach can be used to
obtain a qualitative strengthening (compared to previous proofs) in terms of
"functional error". To illustrate the extra leverage obtained by functional
error, we give an elementary argument for extracting equipartitions from
arbitrary partitions.
