Lebesgue integration

If you are having difficulty understanding this article, you might want to learn about Riemann integrals and the theory of limits first.

In the mathematical branch of real analysis, Lebesgue integration is a framework for extending the notion of integral as the area under the curve to a large class of functions whose domain may not even be in R.

Introduction

The Lebesgue approach is not the most elementary area-based integration theory; that distinction goes to the Riemann integral. The principal advantage of the Lebesgue theory over the Riemann theory is the ease with which limit theorems are proved. Such theorems are needed in the study of Fourier series, Fourier transforms, and elsewhere. It is often also mentioned that a much broader class of functions can be integrated.

Formal construction

Let μ be a (non-negative) measure on a sigma-algebra X over a set E. (In real analysis, E will typically be Euclidean n-space Rⁿ or some Lebesgue measurable subset of it, X will be the sigma-algebra of all Lebesgue measurable subsets of E, and μ will be the Lebesgue measure. In probability and statistics, μ will be a probability measure on a probability space E.) We build up an integral for real-valued functions defined on E as follows.

Fix a set S in X and let f be the function on E whose value is 0 outside of S and 1 inside of S (i.e., f(x) = 1 if x is in S, otherwise f(x) = 0.) This is called the indicating or characteristic function of S and is denoted 1_S.

To assign a value to ∫1_S consistent with the given measure μ, the only reasonable choice is to set:

<math>\int 1_S = \mu (S)</math>

We extend by linearity to the linear span of indicating functions:

<math>\int \sum a_k 1_{S_k} = \sum a_k \mu( S_k)</math>

where the sum is finite and the coefficients a_k are real numbers. Such a finite linear combination of indicating functions is called a simple function. Note that a simple function can be written in many ways as a linear combination of characteristic functions, but the integral will always be the same.

Now the difficulties begin as we attempt to take limits so that we can integrate more general functions. It turns out that the following process works and is most fruitful.

Let f is a non-negative function supported on the set E (we allow it to attain the value +∞, in other words, f takes values in the extended real number line.) We define ∫f to be the supremum of ∫s where s varies over all simple functions which are under f (that is, s(x) ≤ f(x) for all x.) This is analogous to the lower sums of Riemann. However, we will not build an upper sum, and this fact is important in getting a more general class of integrable functions. One can be more explicit and mention the measure and domain of integration:

<math>\int_E f\,d\mu := \sup_{s simple,s \le f} \int_E s\,d\mu</math>

There is the question of whether this definition makes sense (do simple function or indicating function keep the same integral?) There is also the question of whether this corresponds in any way to a Riemann notion of integration. It is not so hard to prove that the answer to both questions is yes.

We have defined ∫f for any non-negative function on E; however for some functions ∫f will be infinite. Furthermore, desirable additive and limit properties of the integral are not satisfied, unless we require that all our functions are measurable, meaning that the pre-image of any interval is in X. We will make this assumption from now on.

To handle signed functions, we need a few more definition. If f is a function of the measurable set E to the reals (including ± ∞), then we can write f = g - h where g(x) = (f(x) if f(x)>0, 0 otherwise) and h(x) = (-f(x) if f(x) < 0, 0 otherwise). Note that both g and h are non-negative functions. Also note that |f| = g + h. If ∫|f| is finite, then f is called Lebesgue integrable. In this case, both ∫g and ∫h are finite, and it makes sense to define ∫f by ∫g - ∫h. It turns out that this definition is the correct one. Complex valued functions can be similarly integrated, by considering the real part and the imaginary part separately.

Theorems

Every reasonable notion of integral needs to be linear and monotone, and the Lebesgue integral is: if f and g are integrable functions and a and b are real numbers, then af + bg is integrable and ∫(af + bg) = a∫f + b∫g; if f ≤ g, then ∫f ≤ ∫g.

Two functions which only differ on a set of μ-measure zero have the same integral, or more precisely: if μ({x : f(x) ≠ g(x)}) = 0, then f is integrable if and only if g is, and in this case ∫ f = ∫ g.

One of the most important advantages that the Lebesgue integral carries over the Riemann integral is the ease with which we can perform limit processes. Three theorems are key here.

The monotone convergence theorem states that if f_k is a sequence of non-negative measurable functions such that f_k(x) ≤ f_k+1(x) for all k, and if f = lim f_k, then ∫f_k converges to ∫f as k goes to infinity. (Note: ∫f may be infinite here.)

Fatou's Lemma[?] states that if f_k is a sequence of non-negative measurable functions and if f = liminf f_k, then ∫f ≤ liminf ∫f_k. (Again, ∫f may be infinite.)

The dominated convergence theorem[?] states that if f_k is a sequence of measurable functions with pointwise limit f, and if there is an integrable function g such that |f_k| ≤ g for all k, then f is integrable and ∫f_k converges to ∫f.