Interactive measure theory

Build area from level sets.

Compare vertical Riemann strips with horizontal Lebesgue bands, then calculate simple functions and test convergence theorems.

Read the 13-page guide LaTeX source

Riemann

partition the domain

Lebesgue

partition the range

Riemann approximation—

Lebesgue approximation—

Difference—

Calculate a simple function.

Each slider sets the height on an interval of length one. Because the intervals are disjoint, the Lebesgue integral is the sum of height times measure.

Heights on three measurable sets

[0,1)3

[1,2)0.5

[2,3]2

Integral

3·1 + 0.5·1 + 2·1 = 5.5

Changing endpoints does not alter the answer: finite sets have measure zero.

Choose the theorem that controls the limit.

Pointwise convergence is only the beginning. Switch sequences to see how monotonicity, domination, or concentration changes what may pass through the integral.

Monotone convergence

Approximate from below.

Nonnegative truncations rise pointwise to the target, so their integrals rise to the integral of the limit.

Nonnegative measurable functions
Increasing pointwise
Infinity is allowed