Integration is the mathematical operation which corresponds to finding the area under a function. A Riemann integral defines this operation by filling the area under the curve with smaller and smaller rectangles. As the rectangles become smaller and smaller, the total area of the rectangles becomes closer and closer to the area under the curve. Unfortunately, there are functions for which this method of finding areas does not work, for example, consider a function f(x) which is 0 when x is rational and 1 otherwise. You can't draw rectangles under the curve and find its area using Riemann integral.
This is where Lebesque integration comes in. Instead of using limits, Lebesque integration uses maximums. Take that function f(x). I don't know the area underneath that function, but I do know that it is greater than all of the functions which are smaller or equal to than the area of all of the functions which are smaller than f(x) across the interval I am interested in. The idea behind Lebesque intergration is to run through all of a set of functions which are smaller or equal to f(x), and the upper bound of those functions is the area of the function.
The main advantage of the Lebesgue integral over the Riemann integral is that more functions become integrable, and that the integral can often be determined easily using convenient convergence theorems. Furthermore, the extension of the notion of integration to functions defined on general measure spaces, which include probability spaces, allows the proper formulation of the foundations of probability and statistics. A formal introduction of the concept follows.
Fix 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 1S.
To assign a value to ∫1S consistent with the given measure m, the only reasonable choice is to set ∫1S:=m(S).
We extend by linearity to the linear span of indicating functions: ∫∑ak1Sk:=∑akm(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 works and is most fruitful.
If f is a non-negative function on 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) and h(x)=0 if x>0 and g(x)=0 and h(x)=-f(x) if f(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 fk is a sequence of non-negative measurable functions such that fk(x)≤fk+1(x) for all k, and if f=lim fk then ∫f_k converges to ∫f as k goes to infinity. Fatou's Lemma states that if fk is a sequence of non-negative functions and if f=limsup fk then ∫f≤liminf∫fk. The Dominated Convergence Theorem states that if fk is a sequence of functions with pointwise limit f, itself an integrable function such that |fk(x)|≤f(x) for all x, then ∫f_k converges to ∫f.