Mathematics / advanced
Measure and Lebesgue integration
Construct integration from measurable sets and functions, then justify limit operations and changes of measure.
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Module 01 advanced
Measure and Lebesgue integration
Construct integration from measurable sets and functions, then justify limit operations and changes of measure.
- 01
Sigma algebras and measurable maps
Specify observable sets and test functions through their inverse images before assigning any sizes.
28 min - 02
Outer measures and the Carathéodory construction
Turn covering costs on arbitrary sets into a complete measure on a precisely defined sigma algebra.
31 min - 03
Lebesgue measure and nonmeasurable sets
Separate topology, cardinality, and measure, then identify the obstruction behind a Vitali set.
30 min - 04
Simple functions and the integral
Build the Lebesgue integral from measurable level sets and distinguish legitimate cancellation from infinity minus infinity.
29 min - 05
Monotone convergence and Fatou’s lemma
Use order to pass through infinite limits and diagnose why mass can disappear from a pointwise limit.
30 min - 06
Dominated convergence and uniform integrability
Control integral limits with one integrable envelope or, on a finite measure space, uniformly small value tails.
32 min - 07
Product measures and Tonelli–Fubini
Construct product integration and decide whether changing the order of integration is justified.
34 min - 08
Signed measures and the Jordan decomposition
Separate positive and negative mass intrinsically and compute total variation without confusing cancellation with absence.
31 min - 09
Radon–Nikodym derivatives
Express one measure as weighted integration against another and identify the singular mass that no density can represent.
33 min - 10
Lebesgue differentiation and absolute continuity
Recover local values from averages and determine exactly when an integrable derivative reconstructs a function.
35 min