# History of cdf as to how it was defined prior Kolmogorov and how the paradigm shifted from $<$ case to $\leq$ case

Cumulative distribution function is pretty much known and well covered from undergrad to grad probability texts. Specifically if $$\mathbf P$$ is a probability measure on $$(\mathbb R, \mathfrak B_\mathbb R),$$ then cdf is defined as

$$\mathsf F(x):= \mathbf P((-\infty, x])\tag 1\label 1.$$

However, early texts like those by Kolmogorov, Uspensky, Gnedenko defined it (borrowing from Kolmogorov's notations) as

$$\mathsf F^{(x)}(a) := \mathbf P(-\infty, a).\tag 2\label 2$$

From $$\eqref 1,~~\forall~a

$$\mathbf P(\{\omega:X(\omega)\in(a, b]\}) = \mathsf F(b)-\mathsf F(a).\tag 3\label 3$$

While from $$\eqref 2,$$

$$\mathbf P\{x\in[a,b)\} = \mathsf F^{(x)}(b)-\mathsf F^{(x)}(a).\tag 4\label 4$$

While $$\eqref 1$$ is right-continuous, $$\eqref 2$$ is left-continuous.

In general too, while I was acquainted with Stieltjes premeasure on Borel field $$\mathfrak B_{(~~~]}(\mathbb R)$$ which is generally defined for a non-decreasing, right continuous $$F:\mathbb R\to\mathbb R$$ as

$$\chi_F(a, b] := F(b)- F(a),\tag 5\label 5$$

in Schilling's measure theory text, the author defines Stieltjes function for monotonically increasing and left-continuous $$F$$ (he calls it Stieltjes function) as

$$\nu_F([a, b)):= F(b)-F(a).\tag 6\label 6$$

My question is why Kolmogorov and his other contemporaries defined the cdf as $$\eqref 2$$ and not as $$\eqref 1$$ and how $$\eqref 1$$ became the more common face as we are accustomed to. Which author first used $$\eqref 1$$ as the definition rather than $$\eqref 2?$$ Was Kolmogorov the first one in history to define cdf as $$\eqref 2?$$ Did Borel, Jordan introduce any formal definition on the same prior to Kolmogorov in their treatises?

In a nutshell, my query revolves around the timeline of the introduction of cdf and its development to the one we are familiar with today.