Recently I have encountered weird notation that I don't see into.

When I have some infinite sum $$\sum_{n=1}^{\infty}f(n)$$ I would rewrite it without thinking to the integral form like this $$\int_{n=1}^{\infty}f(\left \lfloor {x}\right \rfloor)dx$$ where $\left \lfloor {x}\right \rfloor$ is a floor function.

The thing I saw and have difficulty to interpret looked like this $$\int_{n=1}^{\infty}f(x)d\left \lfloor {x}\right \rfloor$$

Thanks for hints and explanations or possibly links to places where I could learn about this.


Ok, thanks for your comments about Stieltjes integral.

So it's just like Riemann integral, except that summation points are chewed through some function (in this case floor function).

$$\int_{n=1}^{\infty}f(x)d g(x)=\sum_{i=1...\infty}f(c_i \in (x_i,x_{i+1}))(g(x_{i+1})-g(x_i))$$

In my case the right bracket $(g(x_{i+1})-g(x_i))$ will be zero everywhere but in integers where the value will be one.


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