Lebesgue integral (thing)

dido's writeup is not entirely correct. The Lebesgue integral is not defined for every extended real function. The function must be measurable. For bounded continuous real valued functions over a set of finite measure, the Riemann integral and Lebesgue integral coincide. The Lebesgue integral of a measurable function may be defined as follows. (Throughout this I shall be concerned only with the Lebesgue measure on R, the real line, which will be denoted by m. If the set over which an integral is taken is not specified, it is assumed to be all of R.)

First we need some definitions.

Let m^* denote the Lebesgue outer measure. A subset E of R is measurable, with respect to the Lebesgue measure iff for any subset A of R

m^*(A) = m^*(A ∩ E) + m^*(A ∩ R\E)

This is the so-called Carathéodory criterion. Since the Lebesgue outer measure is countably sub-additive, we need only require that

m^*(A) ≥ m^*(A ∩ E) + m^*(A ∩ R\E)

An extended real function ƒ:R → R is measurable iff for every extended real number α, the set {x | ƒ(x) > α } is measurable. (This condition is equivalent to the condition that the sets where ƒ(x) ≥ α, ƒ(x) < α, ƒ(x) ≤ α are each measurable.)

Let E be a subset of R, then the characteristic function of E, denoted Χ_E is defined by

Χ_E(x) = 1 if x is in E, 0 otherwise.

An extended real function φ(x) is a simple function iff it assumes only a finite number of values and is measurable (Note: some authors only require that a function assume only a finite number of values in order to be simple; however, for the sake of brevity, I shall only consider so-called measurable simple functions).

If φ(x) is a simple function, then it has a canonical (but not unique) representation

φ(x) = Σ a_iΧ_Ai(x) 1 ≤ i ≤ N

where A_i = { x | φ(x) = a_i}. Note that these are disjoint measurable sets.

For any nonnegative simple function φ(x), with the above canonical representation, and measurable set E we define the Lebesgue integral of φ(x) over E to be

∫_E φ dm = Σ a_im(A_i ∩ E) 1 ≤ i ≤ N

For any nonnegative extended real-valued measurable function ƒ we define the Lebesgue integral of ƒ to be the supremum of all ∫ φ dm, where φ is a simple function such that 0 ≤ φ ≤ ƒ

For a nonnegative extended real-valued measurable function ƒ and a measurable set E, we define the Lebesgue integral of ƒ over E, denoted ∫_E ƒ dm, to be ∫ ƒ*Χ_E dm

A nonnegative extended real-valued measurable function ƒ is (Lebesgue) integrable over a measurable set E iff ∫_E ƒ dm < ∞

If ƒ is an aribitrary extended real-valued function we defined the postive part, denoted ƒ⁺, of ƒ to be

ƒ⁺(x) = max{ƒ(x),0}

Similarly the negative part of ƒ, denoted ƒ^-, is defined as

ƒ^-(x) = max{-ƒ(x),0}

Note that the postive and negative parts of ƒ are both nonnegative extended real-valued functions, and if ƒ is measurable then so are ƒ⁺ and ƒ^-.

An arbitrary measurable function ƒ is integrable over a measurable set E iff ƒ⁺ and ƒ^- are integrable over E, in which case we define the Lebesgue integral of ƒ over E to be

∫_E ƒ dm = ∫_E ƒ⁺ dm - ∫_E ƒ^- dm

Note that ƒ = ƒ⁺ - ƒ^- and |ƒ| = ƒ⁺ + ƒ^- so that ƒ is Lebesgue integrable iff |ƒ| is Lebesgue integrable, which is not the case for Riemann integrable functions.