Concept map · Axler, MIRA, Ch. 1–3
Real analysis
From the limits of the Riemann integral to a theory of measure, and on to an integral that commutes with limits.
- Ch 1. The Riemann integral breaks down. It can’t handle 1ℚ, unbounded functions, or limits.
- Ch 2. This motivates a theory of the size of sets: outer measure, σ-algebras, measurable functions, measures and Lebesgue measure.
- Ch 3. The payoff is a better integral with good limit theorems, one that agrees with Riemann’s wherever Riemann’s works.
Click or tap a box to open its summary. Hover over a box to highlight its prerequisites and consequences. Result numbers such as 2.85 refer to Axler’s book.
Source & attribution
This map follows the structure, section labels and result numbering of Chapters 1–3 of Sheldon Axler’s book. The summaries, key ideas and questions are our own condensed paraphrases, not quotations. Changes from the original: the material is condensed, reorganised into a concept map, and re-expressed in our own words. For the full statements and proofs, read the book: