Two ways to slice the area under a graph
The Riemann integral cuts the domain into thin vertical strips: partition the interval [a,b], pick a height on each strip, and add up width times height. This works beautifully for continuous functions, but it is fragile. The price of admission is that the function must not oscillate too wildly — heights inside a thin strip have to settle down as the strip narrows.
The Lebesgue integral turns the picture sideways: cut the range (the y-axis) into thin horizontal layers instead. For a layer at height around y, ask not “how wide is the strip?” but “how big is the set of x where f(x) is near y?” That size is exactly what Lebesgue measure supplies. We then add up (layer value) times (measure of the set at that value).
The function Riemann cannot integrate
Let D be the Dirichlet function on [0,1]: D(x) = 1 if x is rational, 0 if x is irrational. On any subinterval, no matter how tiny, there are both rationals and irrationals, so the supremum of D there is 1 and the infimum is 0. Every upper Darboux sum equals 1 and every lower Darboux sum equals 0; they never meet. So D is not Riemann integrable.
Lebesgue answers in one line. The rationals are countable, so they have measure zero; the irrationals fill [0,1] up to that null set. Layer-by-layer: the value 1 sits on a set of measure 0, the value 0 sits on a set of measure 1, and the integral is 1·0 + 0·1 = 0. The function is so small (it disagrees with the constant 0 only on a null set) that its integral is simply 0.
Dirichlet D on [0,1]: D(x) = 1 (x rational), 0 (x irrational)
Riemann attempt:
any partition P, any subinterval I:
sup of D on I = 1 (I contains a rational)
inf of D on I = 0 (I contains an irrational)
Upper sum U(D,P) = sum (sup)*length = 1*(1) = 1 for EVERY P
Lower sum L(D,P) = sum (inf)*length = 0*(1) = 0 for EVERY P
upper integral = 1, lower integral = 0, 1 =/= 0 -> NOT Riemann integrable
Lebesgue view (slice the range):
set where D = 1 is Q cap [0,1], countable -> measure 0
set where D = 0 is the irrationals, measure 1
integral = 1 * measure{D=1} + 0 * measure{D=0}
= 1 * 0 + 0 * 1
= 0Simple functions: the bricks we build with
To build the layer-by-layer integral rigorously we need clean building blocks. A characteristic function 1_A takes the value 1 on a measurable set A and 0 elsewhere. A simple function is a finite combination of these: s = c_1·1_{A_1} + … + c_n·1_{A_n}, taking only finitely many values, each on a measurable set. Simple functions are the discrete, staircase-like functions whose integral is obvious: just (value) times (measure) of each piece, summed.
Here is the key structural fact that makes the whole theory go: every non-negative [[measurable-function|measurable function]] is the increasing pointwise limit of simple functions. So if we know how to integrate simple functions, and we know how to take limits well, we can integrate every measurable function. The next four guides do exactly this, one layer at a time.