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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 $\eqref 1,~~\forall~a<b,$

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

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.

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