Riemann vs Stieltjes Integration

Riemann vs Lebesgue Integrals

The blue curve in the 2D view is the function (x^2) which is to be integrated. We are going to see Stieltjes integration of (x^2) with respect to the monotone functions x^3 and e^x (in 2D view and red coloured curves). Riemann integration is nothing but the area between the green line and the x-axis in the 2D view whereas the Stieltjes integration is nothing but the area between the black lines and the corresponding redlines in the 3D view. The area between yellow lines in 3D view is the Stieltjes Integration of (x^2) with the monotone function as identity, which is same as that of the Riemann Integration of (x^2). The two purple curves are examples of a situation where we have chosen a non monotone function (0.2x^5-x^3). Here this function is although non monotone on the whole of real line, it is piecewise monotone. Hence Stieltjes integral can still be calculated in the parts of the real line where (0.2x^5-x^3) is piecewise monotone.