The Lebesgue integral is defined for simple functions as ∫s dμ = Σ aᵢμ(Eᵢ), extended to non-negative measurable functions by monotone supremum, then to general functions via positive and negative parts. This construction unifies and extends Riemann integration.
From your work with simple functions, you know that a simple function takes only finitely many values: s(x) = a₁ on set E₁, a₂ on E₂, ..., aₙ on Eₙ, where these sets partition the domain. You can think of a simple function as a step function where the steps need not be intervals — they can be any measurable sets. The Lebesgue integral of such a function is defined by the obvious formula: ∫s dμ = Σ aᵢ · μ(Eᵢ). Each term is the height of a step times the measure (size) of the set where that step occurs. This is the entire foundation — everything else is a careful limit process built on this base.
To integrate a non-negative measurable function f, you approximate it from below by simple functions. The key idea: for any such f, there is an increasing sequence of simple functions sₙ with sₙ(x) ↑ f(x) pointwise. The integral of f is defined as the supremum of the integrals of all such approximating simple functions: ∫f dμ = sup{∫s dμ : s simple, 0 ≤ s ≤ f}. Because the approximating functions increase to f, and their integrals are already defined, this supremum captures the "total area" under f — even if f is unbounded or has complicated discontinuities. The Monotone Convergence Theorem then guarantees that this limit behaves as expected.
For a general measurable function f (which may be positive, negative, or both), write f = f⁺ - f⁻ where f⁺(x) = max(f(x), 0) is the positive part and f⁻(x) = max(-f(x), 0) is the negative part. Both f⁺ and f⁻ are non-negative, so they are already integrable by the construction above. Then ∫f dμ = ∫f⁺ dμ - ∫f⁻ dμ, provided at least one of these is finite. If both are finite, f is called Lebesgue integrable, and we write f ∈ L¹(μ).
The power of this construction becomes clear when you compare it to Riemann integration. The Riemann integral partitions the *domain* into subintervals and sums widths times heights. The Lebesgue integral partitions the *range* into value bands and sums values times the measure of the preimage. This inversion is why Lebesgue handles badly-discontinuous functions that Riemann cannot: the measure of a set of discontinuities matters, not the structure of those discontinuities as a subset of the x-axis. The construction through simple functions, monotone limits, and positive/negative splitting is the full, rigorous answer to the question: what does it mean to integrate a general measurable function?