Measure Theory 6 | Lebesgue Integral [dark version]

Lebesgue integration gets its full footing by first defining the integral for nonnegative “step” (simple) functions on an abstract measure space, then extending that definition to general nonnegative measurable functions via a supremum over all step functions that sit below them. The key payoff is a well-defined notion of area under a measurable function—independent of how a simple function is represented—built only from the measure μ of sets, even when the underlying space X is completely abstract.

A measure space is described as a triple (X, A, μ): a set X, a sigma-algebra A of measurable subsets, and a measure μ mapping sets in A to values in [0, ∞]. The integration target is a measurable function f: X → ℝ, meaning every preimage f⁻¹(E) of a Borel set E lies in A. The starting point is the characteristic function 1_A of a measurable set A, whose integral is required to match the set’s “volume”: ∫ 1_A dμ = μ(A). From there, simple functions (also called step functions) are introduced as finite linear combinations of characteristic functions: f(x)=∑_{i=1}^n c_i 1_{A_i}, with measurable A_i.

For simple functions, the integral is defined by summing rectangle areas: ∫ f dμ = ∑_{i=1}^n c_i μ(A_i). A subtlety appears immediately when μ(A_i)=∞. If negative coefficients are allowed, expressions like (−2)·∞ become problematic. The solution is to restrict attention to nonnegative simple functions, where all coefficients c_i are ≥ 0. This leads to the set S⁺ of nonnegative simple functions, which behaves like a “half” vector space: addition is allowed, but scaling by negative numbers is not.

Even for simple functions, the representation is not unique—splitting a set A_i into smaller measurable pieces changes the formula without changing the function’s graph. The integral’s definition is designed to be representation-independent, so the total “area” remains the same no matter how the rectangles are partitioned.

With the nonnegative simple case in place, the Lebesgue integral for general nonnegative measurable functions is built using approximation from below. For a measurable f ≥ 0 that may take infinitely many values, one considers all simple functions h ∈ S⁺ such that h(x) ≤ f(x) everywhere. Each such h has a definite integral from the earlier step. Because h is always below f, these integrals provide lower bounds on the desired integral of f. The Lebesgue integral is then defined as the supremum of all these lower bounds:

∫ f dμ = sup{ ∫ h dμ : h ∈ S⁺, 0 ≤ h ≤ f }.

This supremum always exists in [0, ∞] and can be infinite. When the value is finite, f is called μ-integrable. The construction is fundamentally order-driven on the y-axis (the real line), not on X itself: the measure μ only tells how big sets are, while the “approximation by step functions” uses the ordering of real numbers to squeeze f from below. The result is a general Lebesgue integral defined purely from measure and measurability, setting up the later properties that make it powerful.