# continuity of probability distribution

• Dec 26th 2009, 06:58 AM
rough19
continuity of probability distribution
would you help me how I can prove this sentence? please.
• Dec 26th 2009, 07:42 AM
Moo
Hello,

For the last inequality $\displaystyle P(\liminf_n A_n)\leq \liminf_n P(A_n)$ :

Consider $\displaystyle B_N=\bigcap_{n=N}^\infty A_n$
The sequence $\displaystyle (B_N)$ is increasing.
So $\displaystyle P(\lim_N B_N)=\lim_N P(B_N)$ (basic property of a measure/probability)

But $\displaystyle \lim_N B_N=\liminf_n A_n$

So we have $\displaystyle P(\liminf_n A_n)=\lim_N P(B_N)$

But $\displaystyle \forall N,B_N \subseteq A_N$. Thus $\displaystyle P(B_N)\leq P(A_n)$

$\displaystyle \Rightarrow \lim_N P(B_N)=\liminf_N P(B_N) \leq \liminf_n P(A_n)$

Finally, $\displaystyle P(\liminf_n A_n)\leq \liminf P(A_n)$

(note : $\displaystyle \liminf \equiv \underline{\lim}$)

Does this help ?

For the first inequality, consider taking the compliment of the last inequality.
• Dec 26th 2009, 07:49 AM
rough19
Absolutely.
Thank you so much.