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This is very related to this and this but not exactly what I'm looking for. From Wikipedia I know that the following 4 conditions must hold for a function to be a CDF:

Every cumulative distribution function $F_X$ is non-decreasing and right-continuous. Furthermore, the following must hold:

$\lim_{x\to -\infty}F_X(x)=0$ and $\lim_{x\to +\infty}F_X(x)=1$

I know now what must hold but not how to apply/check it. Can someone provide a minimal example of a function that is a CDF with the necessary steps on how to check it? I couldn't find what I'm looking for elsewhere.

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    $\begingroup$ You prove it has the 3 properties above and that $F(x)=\mathbb{P}_X(-\infty,x]$ $\endgroup$ – badatmath May 10 '19 at 19:20
  • $\begingroup$ @badatmath There are 4 properties, so which 3? Also, my question is on how to prove it? What methods do you use? How do you apply them...etc. $\endgroup$ – Does it matter May 10 '19 at 19:22
  • $\begingroup$ What do you mean by "how to check" the four conditions? Those conditions are not phrased in the terminology of probability. Those are just conditions on a function. E.g. to check the two limits, you just evaluate the limits and prove they are $0, 1$ respectively. $\endgroup$ – antkam May 10 '19 at 21:05
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If you prove it's a distribution function for $X$, meaning proving the following $$F_X(x)=\mathbb{P}_X((-\infty,x])=\mathbb{P}(X\in (-\infty,x])=\int1_{(-\infty,x]}(t) \mathbb{P}_X(dt)=\int_\Omega 1_{(-\infty,x]}(X) \mathbb{P}(d\omega).$$ then you are done, since the three other properties follow from continuity of measures and the fact that a measure is monotone and for a probability measure $\mathbb{P}(\Omega)=1$.

If you prove that $F$ is non decreasing: $x\geq y \Rightarrow F(x)\geq F(y)$, right continuous $$\lim_{y\downarrow x}F(y)=F(x)$$ and the two limits at $\pm \infty$, then you showed it is a distribtion function. To prove it is a distribution function for $X$, you need to know something about $X$. Like it's distribution or density or something..

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