Exercise 0.28 (Properties of cumulative distribution function)

As always, we will use arbitrary sequences ${(s_n)}_{n\in \mathbb{N}}$ along the continuous limits $s\to t$ instead of the continuous limits themselves.

1.

By the downwards monotone convergence we obtain: $$\begin{array}{llll}\hfill \underset{n\to \infty }{\lim <!--\; FUNCTION\; APPLICATION\; -->}P(X\le s_n)& =\underset{n\to \infty }{\lim <!--\; FUNCTION\; APPLICATION\; -->}P\left(X\le t\right)-P\left(s_n<X\le t\right)& \hfill & \\ \hfill & =P\left(X\le t\right)-\underset{n\to \infty }{\lim <!--\; FUNCTION\; APPLICATION\; -->}P\left(s_n<X\le t\right)& \hfill & \\ \hfill & =P\left(X\le t\right)-P\left(\bigcap _{n\in \mathbb{N}}\left\{s_n<X\le t\right\}\right)& \hfill & \\ \hfill & =P\left(X\le t\right)-P(\{X=t\})& \hfill & \\ \hfill & =P\left(X<t\right).& \hfill & \end{array}$$

(We have excluded the obvious case where $s_n=t$ eventually, since in that case theorem assumption follows trivially.)

2.

This follows by the previous part of this exercise from $$\begin{array}{llll}\hfill P\left(X\le t\right)-P(\{X=t\})=P\left(X<t\right)⟹P(\{X=t\})& =P\left(X\le t\right)-P\left(X<t\right)& \hfill & \\ \hfill & =P\left(X\le t\right)-\underset{s\to t^{\text{−}}}{\lim <!--\; FUNCTION\; APPLICATION\; -->}F(s).& \hfill & \end{array}$$

3.

If $F$ is continuous, its right and left limits coincide and so $$P(\{X=t\})=P\left(X\le t\right)-P\left(X<t\right)=F(t)-\underset{s\to t^{\text{−}}}{\lim <!--\; FUNCTION\; APPLICATION\; -->}F(s)=0.$$