Lebesgue integral

First a set theoretic definition: Let E be a measurable set (in terms of Lebesgue measure) in Rⁿ. A dissection of this set is a collection C that consists of disjoint, measurable sets whose union is E.

Let f be an extended real function defined on E, and let C be a dissection of E. For every set C_r in C, let

B (f) = sup f(x)
 r      x∈C
           r
b (f) = inf f(x)

 r      x∈C
           r
h (f) = sup f(x) = max(|b (f)|, |B (f)|)
 r      x∈C              r        r
           r

Let us say that if h_r(f) = ∞ and m(C_r) = 0, or vice-versa, then we take h_r(f)m(C_r) = 0. A dissection C such that

Σ h (f)m(C ) < ∞
   r      r

is called admissible. Let the collection of all admissible dissections for a function f over a measurable set E be denoted A(f, E) or just A, if the function and set are understood.

For any dissection C in A, we define the sum

S (f) = Σ B (f) m(C )
 C         r       r

to be the upper approximating sum and similarly

s (f) = Σ b (f) m(C )
 C         r       r

for the lower approximating sum of f over E corresponding to C.

We now define the upper integral with respect to Lebesgue measure as

 *
∫ f = inf S (f)
 E    C∈A  C

and the lower integral as

∫  f = sup s (f)
 *E    C∈A  C

We say that a function f is Lebesgue integrable over a set E if its upper and lower integrals over E are equal, and the value of both is defined to be the Lebesgue integral of f over E.

Clearly, any function is integrable over a set of measure zero, and its integral is zero. Also, any function that is zero almost everywhere (such as the Dirichlet function) is integrable over any set, and its integral is also zero.

A necessary and sufficient condition that f should be integrable over E is that for any ε > 0 there should exist an admissible dissection C of E where S_C - s_C < ε.

There other ways of defining the Lebesgue integral, such as the Daniell integral, which are equivalent.

This is clearly a much more complicated definition of the integral than the Riemann integral, but it has many benefits over that definition, chief of which is that the set of functions integrable in this way forms a Banach space given the natural norm, the Riesz-Fischer Theorem establishes this important result. Also many other functions whose integral does not exist in the Riemann definition have Lebesgue Integrals.

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.