Build the integral from simple functions and see why it dominates Riemann's approach
The Lebesgue integral is the natural integration theory built on measure theory. While the Riemann integral partitions the domain into small intervals and approximates the function on each, the Lebesgue integral partitions the range and measures the size of each preimage. This seemingly simple change of perspective has profound consequences: the Lebesgue integral can handle a vastly larger class of functions and has far better convergence properties.
Henri Lebesgue introduced his integral in his 1902 doctoral thesis, revolutionizing analysis. His approach resolved longstanding problems about the interchange of limits and integrals, provided a natural framework for Fourier analysis, and laid the groundwork for functional analysis and probability theory. Today, the Lebesgue integral is the standard tool throughout mathematics, physics, and engineering.
The construction proceeds in three stages: first for simple functions (where the integral is a finite weighted sum), then for non-negative measurable functions (via suprema of simple function integrals), and finally for general measurable functions (by decomposing into positive and negative parts). Each stage builds naturally on the previous one.
Recall that a simple function φ = Σᵢ₌₁ⁿ aᵢ · χ_{Eᵢ} takes finitely many values a₁, …, aₙ on measurable sets E₁, …, Eₙ. Its integral with respect to a measure μ is defined as ∫ φ dμ = Σᵢ₌₁ⁿ aᵢ · μ(Eᵢ). This is the "obvious" definition: multiply each value by the measure of the set where the function takes that value, and sum.
This definition is well-defined (independent of the particular representation of φ), linear (∫ (αφ + βψ) dμ = α ∫ φ dμ + β ∫ ψ dμ), and monotone (if φ ≤ ψ then ∫ φ dμ ≤ ∫ ψ dμ). These three properties — well-definedness, linearity, and monotonicity — are exactly what we need to extend the integral to all measurable functions.
The key idea for extending beyond simple functions is the approximation theorem: every non-negative measurable function f is the increasing limit of simple functions φₙ ↗ f. We define ∫ f dμ = limₙ ∫ φₙ dμ. The Monotone Convergence Theorem (covered in the next lesson) guarantees this limit is well-defined and independent of the approximating sequence.
The fundamental difference between the two integrals is what gets partitioned. TheRiemann integral divides the domain [a, b] into subintervals [xᵢ, xᵢ₊₁] and forms sums Σ f(xᵢ*) · (xᵢ₊₁ − xᵢ). The Lebesgue integral divides the range of f into thin horizontal strips [yⱼ, yⱼ₊₁) and forms sums Σ yⱼ · μ({x : yⱼ ≤ f(x) < yⱼ₊₁}). The Riemann approach asks "what does f do on this interval?" while the Lebesgue approach asks "where does f take this value?"
Lebesgue himself gave a beautiful analogy: imagine counting a pile of bills. The Riemann approach is like picking up the bills one by one in order and adding their values. The Lebesgue approach is like sorting the bills by denomination first — all the $1 bills, all the $5 bills, etc. — and then multiplying each denomination by its count. Both give the same total, but the sorting approach is more systematic and works even when the pile is wildly disorganized.
Key insight: The Riemann integral requires the function to be "well-behaved" on each subinterval (roughly, continuous almost everywhere). The Lebesgue integral only needs the level sets {f ≥ a} to be measurable. This is why the Lebesgue integral handles far more functions — measurability is a much weaker condition than continuity.
The Dirichlet function D(x) = χ_ℚ(x) equals 1 on the rationals and 0 on the irrationals. It is discontinuous everywhere: in every interval, no matter how small, there are both rationals (where D = 1) and irrationals (where D = 0). This function is the canonical example of a function that is not Riemann integrable.
For any partition of [0, 1], every subinterval contains both rationals and irrationals. The upper Riemann sum is always 1 (choosing rational sample points) and the lower Riemann sum is always 0 (choosing irrational sample points). Since these never converge, D is not Riemann integrable.
But D is Lebesgue integrable! Since ℚ ∩ [0, 1] is measurable with m(ℚ ∩ [0, 1]) = 0, we have ∫₀¹ D dm = 1 · m(ℚ ∩ [0, 1]) + 0 · m((ℝ \ ℚ) ∩ [0, 1]) = 1 · 0 + 0 · 1 = 0. The Lebesgue integral handles D effortlessly because it measures the level sets: the set where D = 1 has measure zero, and the set where D = 0 has measure one. The answer is simply 1 · 0 + 0 · 1 = 0.
Beyond handling more functions, the Lebesgue integral has superior convergence properties. The Monotone Convergence Theorem and Dominated Convergence Theorem provide clean, general conditions under which limₙ ∫ fₙ = ∫ limₙ fₙ. No comparable results exist for the Riemann integral without additional hypotheses like uniform convergence.
The Lebesgue integral also yields complete function spaces. The space L¹(ℝ) of Lebesgue-integrable functions (modulo functions equal almost everywhere) is a Banach space — every Cauchy sequence converges. The analogous space of Riemann-integrable functions is not complete, which makes it unsuitable for the needs of functional analysis, PDE theory, and quantum mechanics.
Furthermore, the Lebesgue integral works naturally on abstract measure spaces, not just on ℝⁿ. This generality is essential for probability theory (where the underlying space can be any sample space), ergodic theory (where we integrate over dynamical systems), and harmonic analysis (where we integrate over locally compact groups). The Riemann integral, tied as it is to the structure of ℝⁿ, cannot match this versatility.