ISEGORIA / MATH ENCYCLOPEDIA
Measure theory: size, approximation, and convergence
Compare partitions, simple functions, and a sequence that defeats dominated convergence.
Before you begin: Real analysis and sets
Predict, manipulate, then check your reasoning against the example and question. Graphs illustrate the mathematics; they do not replace a proof.
1. Two ways to measure area
The same function, two lower sums. Riemann cuts the x-axis into equal pieces and takes the least value on each. Lebesgue cuts the y-axis into levels and adds Δy times the length of each set {f ≥ y}, however scattered it is. Drag the gold level to see one such set on the axis; both sums approach the same integral here.
Worked example. A finer partition can raise a lower simple approximation without changing the function.
Watch out. A picture of rectangles does not prove measurability.
What is being refined?
The information used to assign values and measure the resulting pieces.
2. Simple-function approximation
Raise the level n of the lower simple approximation to the square-root function. Each step of sₙ is constant on the preimage of a value band, marked on the axis, and the dashed previous level always lies below. The filled area increases towards 2/3.
Worked example. The approximations increase pointwise and their integrals approach the target integral.
Watch out. Monotone increase is a property of this construction, not every approximation scheme.
Why does the area never overshoot?
Each step is obtained by rounding the function value downward.
3. A convergence warning
Each fₙ is a spike of height n on (0, 1/n), so its integral is always 1. Drag the point x: the values fₙ(x) become 0 once n ≥ 1/x, so the pointwise limit is 0. The dashed envelope sup fₙ behaves like 1/x and is not integrable, which is why dominated convergence cannot be used.
Worked example. Pointwise convergence alone does not justify exchanging a limit and an integral.
Watch out. The sequence is not dominated by one integrable envelope independent of n.
Which quantity refuses to converge to the integral of the limit?
The integrals remain one although the pointwise limit is zero.