The note follows a simple idea: control a sequence by changing its scale at each point. It develops both a countable-decomposition argument and a proof through dominated convergence.
Background
Measure theory and convergence of measurable functions.
Read the full exposition
The complete argument is available as a PDF.
Revision history ·
Revised September 22, 2026: clarified positive indexing, the convention for empty suprema, and the domain of the measurable envelope. The PDF and TeX source agree.