202405301320
Status: #idea
Tags: Calculus, Integrals
State: #nascient

Riemann-Stieltjes Integrals ~ Generalization of Riemann integrals

We can generalize Riemann Integrals to arbitrary functions instead of always using $x$ (which we remark is a non-decreasing function) as follows:

For $\alpha$ which is non-decreasing. This can be used if $\alpha$ has finite variation, but as I am typing this I am not too sure what that means.

Observe that by our trusty chain rule, the LHS of the above can be written as:

Now note that if $\alpha (x)=x$ then this becomes a Riemann integral. There are cases where $\alpha$ is not differentiable though.

Based on this definition we can define many things such as:

• cumulative density functions
• expected values
• variances

References