If $f$ is a function satisfying-
$_{1)}$ $f(a)\ge 0$, $\forall a\in\mathbb{R}$
$_{2)}$ $\sideset{_{-\infty}}{^\infty}\int f(x)dx =1$
then, $f$ is a density function of the distribution function $F$, $$F(a)=\sideset{_{-\infty}}{^a}\int f(x)dx.$$
I need to verify whether $F(a)$ satisfies all the properties of a distribution function, which are-
$_{1)}$ $0\le F(x)\le 1$, $\forall x\in\mathbb{R}$
[$\because f(x)\ge 0$, we have $\inf\{U(P,f)\}=\sup\{L(P,f)\}\ge 0$, $\forall P.$
Also, it follows from property $_{(2)}$ that $\forall a<\infty$, $F(a)\le F(\infty)=1$]
$_{2)}$ $F$ is non-decreasing
[$\because f$ is non-decreasing and non-negative
$\therefore$ if $a_1\le a_2$, then
$$F(a_1)=\sideset{_{-\infty}}{^{a_1}}\int f(x)dx\le \sideset{_{-\infty}}{^{a_2}}\int f(x)dx=F(a_2)]$$
$_{3)}$ $F(-\infty)=0$ and $F(+\infty)=1$
$_{4)}$ $F(x+)=F(x)$, $\forall x$
- Is the proof for $(1)$ considered correct?
- The justification for $(2)$ seems very intuitive. How should I actually go about the proof or how do I support my claim using other theorems?
- Is it correct to say that $(3)$ follows from $(2)$ and $(1)$?
- How do I prove property $(4)$ is followed?