liminf and limsup of events: complement

Question

Consider a sequence $A_i, i \geq 1$ of Events.

At one point in our lecture, we have used:

$$P[\limsup_i A_i]= 1 -P[\liminf_i A_i^c]$$

with the reasoning that we can use de Morgan's rule.

At some other point however, we write

$$P[\liminf_i A_i] = 1- P[(\liminf_i A_i)^c] = 1- P[\liminf_i A_i^c] \geq 1- P[\limsup_i A_i^c]$$

arguing that $\liminf_i A_i^c \subseteq \limsup_i A_i^c$.

Which is true? Is $(\liminf A_i)^c = \limsup A_i^c$ or equal to $\liminf A_i^c$?

Thanks!

Answer 1

1. $$(\liminf A_i)^c = \limsup A_i^c$$

Pf:

$$LHS = (\liminf A_i)^c = (\bigcup_{m \ge 1} \bigcap_{n \ge m} A_i)^c$$

$$= \bigcap_{m \ge 1} \bigcup_{n \ge m} A_i^c = RHS \ QED$$

2. $$\liminf A_i \subseteq \limsup A_i$$

Pf:

$$\omega \in \liminf A_i$$

$$\to \exists m \ge 1 \ s.t. \omega \in A_m, A_{m+1}, ...$$

Now $\forall n \ge 1,$ we must find $A_i$ s.t. $\omega \in A_i$ and $i > n$

Case 1: $n < m$

Choose $i = m$

Case 2: $n \ge m$

Choose $i = n+1$

QED