0

From the book, Probability and Measure, Billingsley, 3rd, I have problems understanding Eq. 6.6, Page 88:

$$P(\sup_k N_k \leq x ) \leq P(N_n \leq x)$$

where $N_n=1_{A_n}$ and $A_n$ is an iid sequence of events.

More specifically, I have problems with $(\sup_k N_k \leq x )$. I understand that it can be written as $$\{\sup_k N_k \leq x\} = \{\omega : \sup_k N_k(\omega) \leq x\},$$ and I know the definition $$\lim \sup_n (A_n \leq x) = \cap_{n=1}^\infty \cup_{k=n}^\infty (A_k \leq x ),$$ but I don't know how to interpret $(\sup_k N_k \leq x )$.

Thanks in advance!

user3889486
  • 237
  • It's $(\sup_kN_k)\leq x$, not $\sup_k(N_k\leq x)$. – Andreas Blass Oct 28 '17 at 22:04
  • from the book, the inequality is $P(\sup_k N_k \leq x ) \leq P(N_n \leq x)$ – user3889486 Oct 28 '17 at 22:18
  • Yes, and I was telling you what the left side of that inequality means. You said you don't know how to interpret $(\sup_kN_k\leq x)$ and the previous line(about $\limsup$) suggested that you were thinking in terms of the sup of some events, whereas the intended meaning is about the sup of some random variables $N_k$. – Andreas Blass Oct 28 '17 at 23:10

0 Answers0