We have paid the entrance fee. A measure \mu now tells us how big a measurable set is, and a measurable function f is one whose level sets \{f \ge y\} are things \mu can weigh. With those two tools in hand we can finally do the thing the whole module was built for: define the integral

And we define it in a way that will look strange to anyone raised on Riemann. There are no vertical strips, no upper sums, no two-sided squeeze. Instead we build the integral from below, in three clean stages — first for the simplest possible functions, then for every non-negative measurable function by a single supremum, then for signed functions by splitting into a positive and a negative part. Each stage is a few lines. That economy is the whole point: the hard work went into measure theory, and now the payoff arrives almost for free.

Stage 1 — simple functions: integrate by counting values

A simple function is a measurable function that takes only finitely many values. Every such non-negative \varphi can be written in standard form as a finite sum of indicator functions of disjoint measurable sets:

Its integral is the one definition you would guess — value times size, summed:

We adopt the convention 0 \cdot \infty = 0: a height of zero over a set of infinite measure contributes nothing (the graph has no area), and a positive height over a null set contributes nothing either. This makes the arithmetic in [0, \infty] behave.

Well-defined. A simple function has many representations — 2\,\mathbf{1}_{[0,1]} equals 2\,\mathbf{1}_{[0,\frac12)} + 2\,\mathbf{1}_{[\frac12,1]} — so we must check the value \sum c_k \mu(A_k) does not depend on which one we pick. It does not: given two representations, pass to the common refinement formed from all the pairwise intersections A_k \cap B_j; finite additivity of \mu splits each term exactly in step, and the two sums collapse to the same number. From the same refinement trick, integration of simple functions is linear (\int(a\varphi + b\psi) = a\int\varphi + b\int\psi) and monotone (\varphi \le \psi \Rightarrow \int\varphi \le \int\psi).

Stage 2 — non-negative functions: the integral from below

Now let f \ge 0 be any measurable function — possibly wild, possibly unbounded. We define its integral as the supremum of the integrals of all simple functions that fit underneath it:

Read it slowly. We look at every simple staircase \varphi that stays below f everywhere, compute its honest area \int\varphi by Stage 1, and take the least upper bound over all of them. That single number is the integral. It is always defined (the set of candidate values is non-empty — \varphi=0 qualifies — and bounded above in [0,\infty]), and for a simple f it recovers Stage 1, since \varphi = f is the largest admissible staircase.

Why is one-sided enough, when Riemann needed both jaws of a vice to close? Because the two theories slice in different directions. Riemann slices the domain and must pin the height on each strip between an inf and a sup — a genuinely two-sided problem, and the source of every failure. Lebesgue slices the range: a simple \varphi \le f chops the values of f into finitely many levels and asks only for the measure of each super-level set. Refining the levels can only add area under f, never overshoot, so pushing the staircases up toward f converges on the true value with nothing needed from above. The supremum is the integral because the range-slicing can never overstate the case.

Drag the slider to add levels. Each new level cuts a fresh horizontal band; for band at height c_k we measure the set where f reaches that high, and the simple integral \sum_k c_k \cdot \mu(A_k) — read out live — climbs toward the area under f. It only ever rises, and it never crosses the curve. That monotone climb from below is the entire content of Stage 2.

Stage 3 — signed functions: split into positive and negative parts

A general measurable f takes both signs. Break it into two non-negative pieces, the positive and negative parts:

Both f^{+} and f^{-} are non-negative and measurable, so Stage 2 gives each of them an integral in [0,\infty]. We call f integrable (or summable) when both are finite:

and then the integral is the difference — a subtraction of two finite numbers, so no \infty - \infty ever appears:

The equivalence with \int|f| < \infty is immediate: since |f| = f^{+} + f^{-}, linearity on non-negative functions makes \int|f| = \int f^{+} + \int f^{-}, and a sum of two non-negative numbers is finite iff both are. This is why Lebesgue integrability is often stated in one breath as "\int|f| < \infty" — the Lebesgue integral is an absolute integral.

Finally, to integrate over a measurable subset E rather than the whole space, multiply by that set's indicator:

The triumph: \displaystyle\int_0^1 \mathbf{1}_{\mathbb{Q}} \, d\mu = 0

Here is the function that Riemann could not touch — the indicator of the rationals, whose upper and lower Darboux integrals were stuck forever at 1 and 0. Under Lebesgue it is not a pathology at all; it is a simple function, and we integrate it by the Stage 1 formula in a single line.

On [0,1], write \mathbf{1}_{\mathbb{Q}} = 1\cdot\mathbf{1}_{A} + 0\cdot\mathbf{1}_{B} where A = \mathbb{Q}\cap[0,1] and B is the irrationals. The rationals are countable, hence a null set: \mu(A) = 0. Therefore

No partitions to refine, no gap to close, no drama. Where the Riemann machine ground to a halt because every subinterval held both a rational and an irrational, the Lebesgue machine simply weighs the set where the function is 1, finds it has measure zero, and returns 0. This one line is the concrete cash value of the entire module — the promise made on "The Need for Measure", now kept.

The basic properties, and how Riemann fits inside

From the three-stage construction the workhorse rules follow (each proved first for simple functions, then lifted by the supremum). For integrable f, g and scalars a, b:

• Linearity: \displaystyle\int (a f + b g)\, d\mu = a\int f\, d\mu + b\int g\, d\mu.
• Monotonicity: f \le g \Rightarrow \int f\, d\mu \le \int g\, d\mu.
• Triangle inequality: \bigl|\int f\, d\mu\bigr| \le \int |f|\, d\mu.
• Vanishing: \int |f|\, d\mu = 0 \iff f = 0 almost everywhere (i.e. the set \{f \ne 0\} is null).

The last one is worth dwelling on: the integral is blind to what happens on a null set. Change f on the rationals, on any countable set, on any measure-zero set at all, and every integral of f is unchanged. "Almost everywhere" becomes the native dialect of the theory.

That last bullet is the reassurance that nothing was lost: for every honest area you already know how to compute, the two integrals return the identical number. The Lebesgue integral is a strict superset — it agrees with Riemann where Riemann is defined, and keeps going where Riemann quits.