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Lebesgue vs Riemann: Same Answer, More Functions

The Lebesgue integral extends Riemann without contradicting it: every Riemann-integrable function is Lebesgue integrable with the same value, the exact criterion for Riemann is measure-zero discontinuity, and the new theory is complete where the old one was not.

Agreement: same value where both apply

The new integral is genuinely an extension, not a competitor. If f is bounded and Riemann integrable on [a,b], then f is also Lebesgue integrable and the two integrals are equal. Nothing you learned in calculus is overturned: the Lebesgue integral of x^2 on [0,1] is still 1/3. The proof sandwiches f between step functions from below and above (the Darboux sums), applies the convergence theorems to those staircases, and reads off equality of the limits.

The exact criterion: discontinuities of measure zero

Measure theory even answers the old question Riemann could only fumble at: which bounded functions ARE Riemann integrable? The Lebesgue criterion for Riemann integrability is crisp: a bounded function on [a,b] is Riemann integrable if and only if its set of discontinuities has measure zero. Continuity is not required everywhere — only almost everywhere.

Lebesgue criterion:  f bounded on [a,b].
   f is Riemann integrable  <=>  { x : f is discontinuous at x } has measure 0.

Apply it:
  Dirichlet D:  discontinuous at EVERY point of [0,1]
                discontinuity set = [0,1], measure 1 =/= 0
                => NOT Riemann integrable.   (matches Guide 1)

  Thomae's function t(x):  t(p/q)=1/q (lowest terms), t(irrational)=0
                continuous at every irrational, discontinuous on Q
                discontinuity set = Q cap [0,1], measure 0
                => IS Riemann integrable, and integral = 0.

  A step function with finitely many jumps:
                discontinuity set is finite, measure 0
                => Riemann integrable.
The dividing line is measure zero: Dirichlet (discontinuous everywhere) fails; Thomae (discontinuous only on the rationals) passes.

What we gain: completeness

The deepest payoff is structural. The space L1 of Lebesgue-integrable functions is complete: every Cauchy sequence (in the distance integral |f_n - f_m|) converges to a limit that is again in L1. This is the Riesz–Fischer theorem, and more broadly the completeness of the Lp spaces. The analogous space of Riemann-integrable functions is not complete — Cauchy sequences can converge to functions that fall outside the class. Lebesgue's theory is the natural completion.

This is the same upgrade story as completing the rationals to the reals: the analytic machinery only really works once limits stay inside the space. Completeness is precisely what makes L1 and L2 the right homes for Fourier analysis, probability, and the solution of differential equations. The layered integral of Guides 2–3 and the convergence theorems of Guide 4 are what buy you this completeness — that is the whole reason the Lebesgue integral was worth building.