Area Functions and Set Functions

Step 1 · Refining the grid: the squeeze of two walls

Step 2 · Column scanning and the standard parabola model

Step 3 · Area functions, set families, and measure

Step 4 · Axioms, exhaustion approximation, and partitions

Interactive tool: lake-grid refinement squeeze. Drag the slider to cut the coarse grid into finer grids (4×4 → 32×32), toggle the "lower bound" and "upper bound" coloring modes, and watch how the four numbers — lower bound, upper bound, gap, and true area — change.

Why must figures in the plane be represented by point sets?

Interactive tool: the rational-number lake counterexample. Drag the slider to refine the grid and see whether the lower and upper bounds squeeze toward the middle as before.