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What's the big idea?

Every calculus student learns Riemann integration first: chop the x-axis into vertical strips, add up f(xᵢ)·Δx. Lebesgue integration flips it around — chop the y-axis instead, group points by their function value, and weigh each slice by the measure of its preimage: Σ yₖ·m(Eₖ).

Riemann partitions the x-axis	Lebesgue partitions the y-axis
Same-width vertical strips	Same-height horizontal slices
Sum: Σ f(xᵢ) · Δx	Sum: Σ yₖ · m(Eₖ)
Strip color = x-position	Slice color = y-value

Same total area. Completely different philosophy.

Where Riemann breaks: the Dirichlet function

D(x) = 1 if x is rational, 0 otherwise. Every sub-interval of [0,1] contains both a rational and an irrational — so the upper sum is always 1, the lower sum always 0, and the gap never closes. Riemann gives up.

Lebesgue measures the irrationals (1) and the rationals (0), and gets a clean answer: ∫D = 0. This single function motivated a whole field.

Measure zero, yet uncountable: the Cantor set

Remove the middle third of [0, 1], then the middle third of every piece that remains, forever. Row k shows what is left after k removals; the right panel tracks the surviving measure. m(E_k) = (2/3)^k collapses to 0 exponentially — yet the limiting set has as many points as the real line.

The Cantor function is the staircase this process grows into: continuous, non-decreasing, flat almost everywhere (F' = 0 a.e.), yet it climbs from 0 to 1 — the counterexample that kills the naive fundamental theorem of calculus.

Every measurable function is a limit of simple functions

A simple function takes finitely many values, each constant on a measurable set. The classic construction cuts the range of f into 2^n equal horizontal bands and snaps every point down to the lower edge of its band: sₙ ≤ f, and sₙ climbs to f as n grows. This is the Lebesgue picture finished — the integral is defined as the limit ∫f = lim ∫sₙ.

Left: f with sₙ staircased beneath it — each band is a level set E_k = {x : c_k ≤ f(x) < c_{k+1}}. Top right: ∫s_k rises monotonically to ∫f. Bottom right: the uniform error ‖f − s_k‖∞ → 0 on a log scale. Simple functions don't care about discontinuities, so oscillations and jumps are handled just as naturally:

A jump discontinuity changes nothing — the level sets stay measurable, so ∫s_n still converges to the true ∫f = 3/4:

Four ways to converge: one family, four cameras

A sequence of functions can converge in genuinely different senses, and the textbook separates them with three counterexamples. convergence_modes draws each family once and judges it under four cameras at the same time — pointwise, uniform, almost-everywhere and L¹ — so the hierarchy breaks apart in front of you.

The x^n family on [0, 1] converges pointwise and almost everywhere (only x = 1 is exceptional) and in L¹ — but not uniformly: the ε-tube around the limit never closes near x = 1.

The thin spike n·1_[0, 1/n] collapses onto the single point x = 0 — a null set — so it converges almost everywhere, yet ∫f_n = 1 stays put: a.e. but not in L¹.

The typewriter sequence marches an indicator interval across [0, 1], its width shrinking as it goes. The integrals ∫f_n → 0, so it converges in L¹ — yet every point is covered infinitely often: it fails at every single point.

All four panels share one grid, one limit f and one value of ε, and the verdicts are plain NumPy. Pointwise and a.e. use the same honest finite-window test — the measure of the set where the last few members still disagree — which is exactly what separates the spike from the typewriter.

Measure breaks down: the Vitali set

Measure theory's sharpest object is a set that no measure can assign a size to. Partition the real line into rational cosets x + ℚ — equivalence classes under x ~ y ⟺ x − y ∈ ℚ, each countable and dense. V is a transversal: the Axiom of Choice picks one point out of every class.

Shift V by each rational r ∈ ℚ ∩ [0,1). These countably many copies V + r partition the circle [0,1), so countable additivity demands 1 = m([0,1)) = Σ m(V + r), while translation invariance forces every term to be the same number m(V) = δ. Hence 1 = Σ_ℵ₀ δ — impossible: the sum is 0 if δ = 0, and infinite if δ > 0. No value of δ works, so V is not Lebesgue measurable. (Honest, stated not derived: the inner measure is m_*(V) = 0 and the outer measure satisfies m^*(V) > 0 — it can even be chosen arbitrarily small, not 1.)

The figure reads as a four-panel proof chain: classes are dense clouds (every window meets every class), Choice assembles V one point per class, the translates tile the circle disjointly — verified numerically, 96 points pairwise distinct mod 1 — and the measure trap snaps shut.

Install

pip install realviz

Quick start

from realviz import compare_integrals, riemann_plot, lebesgue_plot

# Side-by-side comparison — the main event
compare_integrals(lambda x: x**2, 0, 1, n_partitions=15)

# Or inspect each method alone
riemann_plot(lambda x: x**2, 0, 1, n_partitions=10, method="midpoint")
lebesgue_plot(lambda x: x**2, 0, 1, n_partitions=10)

# The Dirichlet function — where Riemann fails, Lebesgue succeeds
from realviz import dirichlet_illustration
dirichlet_illustration()

# The Cantor set and the devil's staircase
from realviz import cantor_set, cantor_function
cantor_set(n_levels=4)     # construction + measure decay
cantor_function()          # F' = 0 a.e., yet F(1) - F(0) = 1

# Simple-function approximation — the Lebesgue story, finished
from realviz import simple_approximation
simple_approximation(lambda x: x**2, 0, 1, n_levels=5)   # s_n -> f, ∫s_n -> ∫f

# Convergence modes — one family, four cameras
from realviz import convergence_modes, FAMILIES
convergence_modes("x_pow_n")     # ✓ pointwise & a.e., ✗ uniform
convergence_modes("thin_spike")  # ✓ a.e., ✗ L1
convergence_modes("typewriter")  # ✓ L1, fails at every point

# The Vitali set — why a transversal of Q has no measure
from realviz import vitali_illustration
vitali_illustration()

Try it yourself

python examples/demo_compare.py     # x² and sin(x), side by side
python examples/demo_dirichlet.py   # the Dirichlet illustration
python examples/demo_cantor.py      # Cantor set + devil's staircase
python examples/demo_simple_approx.py  # simple functions: s_n -> f
python examples/demo_convergence.py    # the three counterexamples, 2x2 grids
python examples/demo_vitali.py         # the Vitali set — a 4-panel proof

Design principles

• Safe by default — strict input validation: functions are sampled defensively, partition counts are bounded, no eval/exec/pickle, no network, no file I/O.
• Pedagogical color — every strip is colored by what is being partitioned, so the axis-choice is visible at a glance.
• Zero magic — pure NumPy + Matplotlib, ~2,400 lines of core code, MIT licensed.

Roadmap

• v0.1.0 — Riemann vs Lebesgue integral comparison + Dirichlet illustration
• v0.2.0 — Cantor set construction + Cantor function (devil's staircase)
• v0.3.0 — Simple-function approximation of measurable functions
• v0.4.0 — Convergence modes (pointwise, uniform, a.e., in L¹)
• v0.5.0 — Vitali set and non-measurable sets (conceptual)

Development

git clone https://github.com/HunterZ549/realviz.git
cd realviz
pip install -e ".[dev]"
pytest               # 230 tests, CI runs Python 3.9–3.12

License

MIT — see LICENSE.