Riemann-Stieltjes integral over a degenerate interval.

On the Wikipedia article of Riemann-Stieltjes integration they require that $a<b$ if we integrate over the interval $[a,b]$. However, in Rudin's "Principles of mathematical" analysis they do not require this.

https://en.wikipedia.org/wiki/Riemann%E2%80%93Stieltjes_integral

https://www.amazon.com/Principles-Mathematical-Analysis-International-Mathematics/dp/007054235X

From what I see the integral $\int_a^a f dF(x)$ is always zero when viewed as a Riemann-Stieltjes integral?

So is this statement true?

Assume you have a random variable X. Define F(x) to be $P(X\le x)$,this function is bounded, monotone, non-deacreasing and right-countinuous and $\lim_{x \rightarrow -\infty} =0$. Define $\mu$ to be theLebesgue-Stieltjes measure of the distribution function $F$. Assume that $P(X=c)>0$.Let $\int_c^c 1\cdot dF(x)$ be the Riemann-Stieltjes integral. We then have

$$\int_c^c 1\cdot dF(x)=0\ne P(X=c)=\int_{[c,c]}1\cdot d\mu.$$