Measure and Lebesgue integrals
Lebesgue measure is length for complicated sets: countable sets and rational points, Cantor's dust, and the Lebesgue integral of a function defined differently on rationals and irrationals.
How to solve it
Lebesgue measure generalizes length: an interval's measure is its length, and a union of disjoint pieces has the sum of their measures. Every countable set — all the rationals, for example — has measure zero: each point can be covered by an arbitrarily short interval. So the Lebesgue integral does not care what happens on a set of measure zero.
Step by step
- Split the set into simple pieces — intervals, points, countable sets — and add the measures. Count overlaps once.
- A countable set (finite, the naturals, the rationals, the rational points of an interval) has measure $0$.
- A “dust” construction: each step keeps the same share. After $n$ steps the measure is that share to the power $n$.
- The Lebesgue integral: if a function equals an ordinary one almost everywhere (except on a set of measure zero), the integral is the ordinary integral of that one.
Common mistakes
- Thinking the rational points of an interval are “many” because they are everywhere, and giving them positive measure.
- In a dust construction removing parts only at the first step: the remaining share multiplies at every step.
- Counting the function's values at the rational points in a Lebesgue integral.
- Adding the measures of overlapping sets without removing the common part.