Lotus BlogTheory - [T15]
Riemman Integral
The Riemman Integral is the most known method to define the integral of a function, due to its intuitive definition. This integral is used to calculate the area under a curve, this is done by dividing the area of the function in question in an increasing number of rectangles and to calculate the sum of the area of these rectangles.
To give a more formal definition let $n$ be the number of rectangles that divide the area underneath a function $f(x)$ and let $dx$ be defined as the width of the base of each rectangle on the $x$ axis. Then we can define the Riemman integral as $$ \lim_{n\to\infty} \sum f(x) dx \implies \int f(x) dx $$
Limitations
Unfortunately although the Riemman integral is very intuitive it only works if the function $f(x)$ is continuous along its domain (along the $x$ axis). This is a big limitation for Probability Theory in which we have to deal with discrete models
Lebesgue-Stieltjes Integral
Lebesgue-Stieltjes Integral overcomes the limitations of the Riemman integral and unifies the discrete and continuous models under a single notation. This new integral extends the Riemman definition for the integral by defining it on a measure space. This is achievable if instead of dissecting the area along the domain we perform the division in rectangles horizontally.
This allows for more flexible calculation of the area under the curve.
As we can see from the image above the area in question doesn’t have a single base, but rather its width is determined by the union of those three intervals ($E_i$). We can immediately understand how this would translate on a discrete case.