How does $t \rightarrow \infty$ then $t[1-F(t)+F(-t)] \rightarrow 0$ relate to the weak law of large numbers? Refering to the notes here http://www.stat.umn.edu/geyer/8112/notes/weaklaw.pdf
In Theorem 1, I understand how (i) $\iff$ (iii). I also understand the second part of (ii) where  $\lim_{t \to\infty} \int ^t _{-t} x F \{dx \} = \mu $
However, I do not understand how the first part of (ii) is of any relevance here. Doesn't that equation hold true for all distribution functions, since as $t \rightarrow \infty$, $F(t) \rightarrow 1$ and $F(-t) \rightarrow 0$?
Therefore, $1-F(t)+F(-t) \rightarrow 0$, and therefore $t(1-F(t)+F(-t)) \rightarrow 0$?
How does this relate to the weak law of large numbers?


$1-F(t)+F(-t) \rightarrow 0$, and therefore $t(1-F(t)+F(-t)) \rightarrow 0$.

If for a function $h\colon\mathbb R\to\mathbb R$, we have $h(t)\to 0$, it does not mean that $t\cdot h(t)$ goes to zero (for example if $h(t)=\sqrt{1+\left\lvert t\right\rvert}$). 
Here, we have
$$1-F(t)+F(-t)=\mathbb P\left\{X_1 \gt t \right\}  + \mathbb P\left\{X_1 \leqslant -t \right\}=\mathbb P\left\{\left\lvert  X_1 \right\rvert  \gt t \right\} +\mathbb P\left\{X_1 =-t  \right\}           $$
hence condition (1a) in the notes can be interpreted as a decay condition on the tail of $\left\lvert  X_1 \right\rvert$.
