RefacteThe mechanism
Change the scale at each point
Pointwise information need not give uniform control. This argument introduces a positive measurable weight that adjusts the scale from point to point. The weighted sequence can then be controlled using a decomposition argument or dominated convergence.
What becomes elementary: Choosing the weight turns local convergence information into a statement about the measure of exceptional sets.
Compare the two proofs in the study. Track exactly where positivity and measurability of the weight are used, and how the conclusion concerns the weighted sequence.
For another construction, explore the mechanism calculi. Edition dates, sources and reference chapters identify the scope of each study.
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.