Lebesgue integration

Let m be a (non-negative) measure on a sigma algebra X over a set E. We build up an integral as follows.

Fix S in E and let f be a function of 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 function of S and is denoted 1_S(x) or simply 1_S.

To assign a value to ∫1_S consistent with the given measure m, the only reasonable choice is to set ∫1_S:=m(S).

We extend by linearity to the linear span of indicating functions: ∫∑a_k1_Sk:=∑a_km(Sk) (where the sum is finite.) Such a finite linear combination of indicating functions is called a simple function.

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 functions and is most fruitful.

If f is a non-negative function of E then 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.

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. Another interesting question is whether all functions f are Lebesgue integrable, in that desirable additive and limit properties hold. The sad answer is no, which is easy to see if one knows about [non-measurable sets]?. However, if we restrict our attention to measurable functions (functions such that the pre-image of any interval is m-measurable) then we can proceed fearlessly.

To handle signed functions, we need a few more definition. If f is a function of E to the reals, then we can write f=g-h where g(x)=f(x) if x>0 else g(x)=0 and h(x)=-f(x) if f(x)<0 else h(x)=0. Note that both g and h are non-negative functions. Also note that |f|=g+h. If ∫|f| is finite (such a function is called integrable), that is, both ∫g and ∫h are finite, then it would make sense to define ∫f by (int;g)-(∫h). It turns out that this definition is the correct one. Complex values functions can be similarly integrated.

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. [Fatou's Lemma]? states that if f_k is a sequence of non-negative functions and if f=limsup f_k then ∫f≤liminf∫f_k. The [Dominated Convergence Theorem]? states that if f_k is a sequence of functions with pointwise limit f, itself an integrable function such that |f_k(x)|≤f(x) for all x, then ∫f_k converges to ∫f.

See null set, [Henri Leon Lebesgue]?,integration,measure

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