# Finite conditional expectations, measurable densities, and positive kernel integrals

Read the lesson’s browser page, or the editable Markdown source. READING.md is the complete editable argument with eighteen tagged equations and five fully solved exercises worth 56 points. The three figures have their full original captions and downloadable PNG/SVG files.

The opening links complete preceding measure-theory proofs. The reading addresses finite upward convergence, bounded-kernel densities with fibrewise absolute continuity for every parameter, and positive s-finite integration on ordinary product sigma-algebras. It also constructs Borel probability codes and identifies probability kernels with measurable maps into the evaluation space. It does not assert a parameter-independent target null set, signed interchange or uniqueness of an s-finite product measure.

To regenerate figures in a fresh directory, copy reproduce.py and fonts, install requirements.txt in an isolated environment, and run `python reproduce.py`. The script writes two figures in PNG and SVG, together with finite-example calculations and font information. Source values, fractions and bounds are exact; examples illustrate the proved general results.

The complete editable lesson is READING.md. Keep its relative figure, font and reproduction paths together. The course reader supplies the stylesheet and mathematics renderer. See COMPONENT-TERMS.md for the component-specific terms and requirements.txt for figure-software versions.

The third figure, probability-codes, is regenerated by ../reproduction/residual-mathematics/draw_figures.py. That script uses the existing bundled DejaVu Sans font and retained Pillow notice; its README and component terms identify the relative paths.