A function the Riemann integral cannot handle
Recall how Riemann integration works: we slice the domain into small intervals, on each piece estimate the function from above and below, and ask whether the upper Darboux sum and lower Darboux sum squeeze together. This works beautifully for continuous functions. It fails spectacularly for functions that wiggle too much.
The classic offender is the Dirichlet function D, equal to 1 on the rationals and 0 on the irrationals. On any subinterval, no matter how tiny, there are both rationals and irrationals, so the supremum of D is 1 and the infimum is 0. The upper sum is always 1 and the lower sum is always 0; they never meet. By the Riemann criterion, D is not Riemann integrable — and yet morally its “area” ought to be 0, because the rationals are a vanishingly small, countable scattering of points.
Limits and integrals refuse to commute
The deeper defect appears with limits. Enumerate the rationals in [0,1] as q_1, q_2, q_3, … and let f_n be 1 on the first n of them and 0 elsewhere. Each f_n is zero except at finitely many points, so each f_n is Riemann integrable with integral 0. But pointwise f_n converges to the Dirichlet function D, which is not integrable at all. The limit of integrable functions need not be integrable — the theory is not closed under the limits we most want to take.
Setup: list rationals in [0,1] as q_1, q_2, q_3, ...
f_n(x) = 1 if x in {q_1, ..., q_n}
= 0 otherwise
Each f_n is 0 except at n points, hence Riemann integrable:
integral_0^1 f_n = 0 (for every n)
Pointwise limit:
for x rational : x = q_k for some k, so f_n(x) = 1 once n >= k -> 1
for x irrational: f_n(x) = 0 for all n -> 0
Thus f_n -> D pointwise, where D = Dirichlet function.
Now compare:
lim_n integral_0^1 f_n = lim_n 0 = 0 (exists)
integral_0^1 ( lim_n f_n ) = integral_0^1 D = DOES NOT EXIST (Riemann)
Conclusion: under Riemann's definition we cannot even ASK whether
lim integral = integral lim ,
because the right-hand side is undefined. The limit escaped the theory.This interchange-of-limits problem is not a curiosity — it is exactly what we constantly need in analysis, probability, and PDE. We want robust theorems that say “if f_n → f and the f_n are not too wild, then ∫f_n → ∫f.” The Riemann integral gives such theorems only under heavy extra hypotheses like uniform convergence. The Lebesgue theory will give them under far gentler conditions.
The change of strategy: slice the range, measure the sets
Riemann slices the domain (the x-axis) into intervals. Lebesgue's idea is to slice the range (the y-axis) instead: ask “for which x is the function value between 0.3 and 0.4?”, then weigh that set of x's. To carry this out we must be able to assign a size — a measure — to the set {x : 0.3 ≤ f(x) ≤ 0.4}, and such a set can be far more complicated than an interval.