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

Pseudofinite proofs of the stable graph regularity lemma