Measure
Can every set be assigned a size?
Length is obvious for an interval and meaningless for a scattering of points. Measure theory is the attempt to extend length as far as it will go — and the discovery that it does not go all the way, which is a theorem rather than a failure.
Outer measure
How do you measure a set with no edges?
Covering a set with intervals
Length is defined for intervals, so measure anything else by covering it with intervals and taking the cheapest cover. The film shows what the cheapest cover misses.
The set that cannot be measured
Not every set has a length. The construction requires the axiom of choice, which is why the theorem is a statement about mathematics as much as about sets.
The Lebesgue integral
Why rebuild an integral that already works?
Slicing horizontally instead of vertically
Riemann sums fail on functions that oscillate too fast. Turning the rectangles on their side fixes it, and the film makes clear why that is more than a trick.
When can you swap a limit and an integral?
The convergence theorems are the reason anyone builds this machinery, and each one is stated with an example showing what happens without its hypothesis.
Measurable functions
Which functions is this integral even defined for?
Almost everywhere, and why it is enough
Two functions differing on a set of measure zero are the same for every purpose here. That equivalence does more work than any theorem in the subject.
Product measures
Does Fubini's theorem ever actually fail?
Measure, as one volume
Every document in this subject is a chapter of the same book, compiled from one source with live cross-references and continuous numbering. Download the whole thing, or take chapters as you go — the page numbers and theorem references agree either way.
Download Volume V ↓