Prerequisites and supporting reading

The following proof passages and definitions are available in the programme’s Measure and Integration reader (D10), in Bahasa Indonesia. The two lessons here are in English. Result numbers agree with Fremlin’s original; the English source references are included in each lesson.

The lifting theorem

• D10 / Fremlin 211A — full programme text (Bahasa Indonesia): Complete measure spaces: every subset of a null measurable set is measurable.
• D10 / Fremlin 211E — full programme text (Bahasa Indonesia): Strict localizability: a partition into finite-measure measurable pieces determines measurability and the measure by arbitrary nonnegative sums.
• D10 / Fremlin 233D — full programme text (Bahasa Indonesia): On a probability space, conditional expectations of real integrable functions exist for every sub-sigma-algebra; the proof and measurable-version discussion are in 233Da–233Dc.
• D10 / Fremlin 233E — full programme text (Bahasa Indonesia): Conditional expectation is linear and order preserving, including bounds 0 <= E(1_G | T) <= 1; the corresponding proof is included.
• D10 / Fremlin 275I — full programme text (Bahasa Indonesia): For increasing sub-sigma-algebras, E(X | Sigma_n) converges almost everywhere and in L1 to E(X | sigma(union Sigma_n)), for real integrable X on a probability space.
• D10 / Fremlin 223B — full programme text (Bahasa Indonesia): Lebesgue density on the real line: symmetric interval densities are 1 almost everywhere in a measurable set and 0 almost everywhere outside it.

Besicovitch's covering theorem and the differentiation of Radon measures

• D10 / Fremlin 134A — full programme text (Bahasa Indonesia): Lebesgue outer measure and completed Lebesgue measure on R^r are translation invariant, with preservation of measurability.
• D10 / Fremlin 263A — full programme text (Bahasa Indonesia): Linear change of variables on R^r: measure of T(E) equals |det T| times measure of E; the invertible integral transformation is also proved.
• D10 / Fremlin 256A — full programme text (Bahasa Indonesia): The Euclidean Radon convention is a complete topological measure, finite on bounded sets and inner regular on all measurable sets by compact subsets.
• D10 / Fremlin 256B — full programme text (Bahasa Indonesia): Every measurable set for such a Radon measure admits closed inner and open outer approximations with arbitrarily small difference; sigma-finiteness and auxiliary facts are proved.
• D10 / Fremlin 225A — full programme text (Bahasa Indonesia): Absolute continuity of the integral of an integrable real function on an arbitrary measure space, in the stronger finite-support formulation.

The lifting lesson explicitly assumes Zorn’s lemma. Its Lemma 3.1 applies 275I only to indicator functions on a probability space and an increasing sequence of sub-sigma-algebras, exactly the theorem’s hypotheses. The sigma-algebra generated by their union is supplied by its own Lemma 1.3. Conditional-expectation linearity and order are proved in 233E, supplementing existence and measurable versions in 233D.

For the density phenomenon in lifting Exercise 7.3, there is also an English proof of the general Radon density conclusion in the companion Besicovitch lesson, Corollary 5.2. The exact Lebesgue-on-the-line formulation used by that exercise remains linked above as D10 223B.

The lifting proof follows Fremlin Volume 3, §341; the covering and differentiation proofs follow Volume 4, §472. Both full proofs are written in the selected lessons themselves. The vector-valued-functions lesson named under “Where this leads” is a later consumer, not an incoming proof dependency, and is not included in this edition.

The programme book also contains the earlier proofs used by these passages, including Radon–Nikodym (232E), measurable versions (232He), martingale convergence (275G–275H) and the monotone-class theorem (136B). Follow the theorem links within the reader to study these prerequisites.