Properties of a distribution function

Question

I'm having trouble understanding the properties of a distribution function. My book only gives these short rules.

http://www.pixhost.org/show/2720/28297379_2015-06-22-15-27-44.jpg

My professor said also said any c.d.f. (Cumulative distribution function) must be right-continuous everywhere.

Can someone please explain these rules to me. I've been trying to google and read my textbook for more explanation but I have not been able to find a good explanation on these rules. For my hw I need to be able to prove these rules.

Answer 1

A function $f:\mathbb R\to \mathbb R$ is right-continuous if $\forall x\in\mathbb R,\ \lim_{t\to x^+}f(t)=f(x)$.

The c.d.f. of a probability $\mathbb P$ is

$$F_{\mathbb P}(x):=\mathbb P\left((-\infty,x]\right)$$

For $\sigma$-additive probabilities over a $\sigma$-algebra $\mathcal E$ the following properties hold:

a. $A\subseteq B\Longrightarrow \mathbb P(A)\le\mathbb P(B)$

b. $A_n\uparrow A\in \mathcal E\Longrightarrow \mathbb P(A)=\sup_n\mathbb P(A_n)$

c. $A_n\downarrow A\in \mathcal E\Longrightarrow \mathbb P(A)=\inf_n\mathbb P(A_n)$

(3) follows immediately from (a)

(1) follows from (c) + (3), noticing that $(-\infty,n]\uparrow \mathbb R$

(2) + right-continuity follows from (b) + (c), noticing that $(-\infty,-n]\downarrow \emptyset$ and $\left(-\infty, x+\frac1n\right]\downarrow (-\infty,x]$