Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

D.3 LEBESGUE INTEGRAL

For function y = g(x), a particular value of x is mapped only to one value of y; this, of course, is the defining property of a function. If the function is not one-to-one, then many values of x could map to the same y, such that the inverse image g−1(y) is one-to-many. This distinction is needed when initially describing the Lebesgue integral, which is obtained by partitioning the y-axis instead of the x-axis (as is done for the Riemann integral). Partitioning the y-axis allows integrals to be computed for functions that are not Riemann integrable, where the x-axis cannot be partitioned in a useful manner.

Consider y = g(x) shown in Figure D.3(a) where the y-axis in the interval [g(a), g(b)] has been partitioned into subintervals {[yn, yn + 1)}. Observe that this yields horizontal rectangles whose ends are constrained by the function. We would like to compute for each yn the area of all vertical rectangles formed by subintervals on the x-axis with height given by yn. For the example in Figure D.3(b), we see there is one vertical rectangle with height g(a) (denoted by the diagonal lines). There are three vertical rectangles (with the darkest shading) whose heights are given by the next y value in the partition, and then two rectangles for the next y value. Multiple rectangles occur because the inverse image from y to x is not one-to-one. Each of the other values in the partition of the y-axis corresponds to a single vertical rectangle (with the lightest shading). ...