# Riemann-Stieltjes integral of a general function f with respect to the greatest integer function.

I want to prove $$\sum_{n\leq x} f(n)= \int_{y}^{x}f(t)d[t]$$ where the interval of integration is [y,x]. Here n is an integer. My attempt:- By Riemann Stieltjes integration's definition, we have $$\int_{y}^{x}f(t)dt = \lim_{n \to +\infty}\sum_{ k=1}^{n-1} f(c_{k})([x_{k+1}]-[x_{k}])$$ where is $$c_{k}$$ is in $$[x_{k}, x_{k+1}]$$. I do not know how to proceed further. Kindly guide me further.

I will assume that $$f$$ is a continuous function on the interval $$[a, b].$$ Consider a partition $$P = (x_0 = a < x_1 < \dots < x_{n - 1} < x_n)$$ and a system of intermediate points $$\xi = (\xi_1, \dots, \xi_n).$$ The the associated Riemann-Stieltjes sum is simply $$S(f; g, P, \xi) := \sum_{1 \leq i \leq n} f(\xi_i)(g(x_i) - g(x_{i - 1})) = \sum_{1 \leq i \leq n} f(\xi_i) (\lfloor x_i \rfloor - \lfloor x_{i - 1} \rfloor).$$ We consider a sequence of partitions $$P_n = (x_0^n < \dots < x_N^n)$$ with $$\Vert P_n \Vert \to 0$$ and any associated system of intermediate points $$\xi_n.$$ It follows that for every integer $$k$$ between $$a$$ and $$b$$ there is a unique interval $$[x^n_{i_k - 1}, x^n_{i_k}]$$ such that $$k \in [x^n_{i_k - 1}, x^n_{i_k}]$$ and $$\lfloor x^n_{i_k} \rfloor - \lfloor x^n_{i_kk - 1} \rfloor = 1$$ for $$n$$ sufficiently large (if $$n$$ happens to be one of the ends of this interval, then it would be found in two intervals of this type; however, there is only one that also verifies the second condition since $$\Vert P_n \Vert$$). Then $$S(f;g, P_n, \xi_n) = \sum_{1 \leq i \leq N} f(\xi^n_i) (\lfloor x^n_i \rfloor - \lfloor x^n_{i - 1} \rfloor) = \sum_{1 \leq i \leq N} f(\xi^n_{i_k}).$$ Since $$\xi^n_{i_k}$$ is always in the interval that contains $$k,$$ $$\Vert P_n \Vert \to 0$$ and $$f$$ is continuous, we get that $$f(\xi_{i_k}^n) \to f(k)$$ as $$n \to \infty.$$ It follows that $$\int_a^b f d\lfloor t \rfloor = \sum_{n \in \mathbb{Z} \cap [a, b]} f(n),$$ where this sum is taken to be $$0$$ if the intersection is empty. I hope this helps. :)
• I think you meant $f(\xi_{i_{k}}^{n}) \rightarrow f(k)$ as n tends to infinity instead of just k. Commented Feb 18, 2021 at 3:07