The Lebesgue Integral and What It Repairs
Define the integral first for simple functions, then as a supremum over the simple functions underneath. Functions the Riemann theory cannot touch become integrable, and the value agrees with the Riemann integral wherever that one exists.
What a learner can do afterwards
- Integrate a simple function directly from its defining partition
- Integrate the indicator function of the rationals and say where the Riemann definition stalls
- State how the two integrals relate on a bounded interval
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A simple function is constant on each piece of a finite measurable partition. Its integral is the sum over pieces of value times measure, and a piece of measure zero adds nothing. For a non-negative measurable f, the integral is the supremum of these sums over simple functions below f, so the staircases from the last stop climb to its value.
The indicator of the rationals is 1 on rationals and 0 elsewhere. On [0, 1] its integral is 0, because the rationals have measure zero and the irrationals carry the full measure. Riemann sums stall here: every subinterval holds both kinds of points, so lower sums sit at 0 while upper sums sit at 1.
Riemann cuts the domain into subintervals, so wild oscillation inside one piece ruins the sums. Lebesgue cuts the range into levels and measures how much of the domain sits at each height. Grouping by height is what tames functions like the rationals indicator.
On a bounded interval, every Riemann integrable function is Lebesgue integrable, with the same value. The new integral agrees with the old one wherever the old one works, and it also handles functions the old one cannot touch.
The Lebesgue integral groups a function by height levels instead of domain slices, so it handles wilder functions and still agrees with Riemann where Riemann works.
Lesson, questions and flashcards are original text written for this dataset.
Reading notes and questions are original text written for this dataset, and flashcards were drafted from collected fact candidates.
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