3 of 8 Improving LaTeX readability.

"this class of problems lies" in RE, so its name is "RE".

$$\begin{align*}\operatorname{Prob}(M \text{ accepts}) &= \operatorname{Prob}\big((\exists n)(M \text{ accepts after exactly $n$ steps})\big)\\ &=\sum_n \operatorname{Prob}(M \text{ accepts after exactly $n$ steps})\\ &=\lim_N \sum_{n\leq N} \operatorname{Prob}(M \text{ accepts after exactly $n$ steps})\\ &=\lim_N \;\operatorname{Prob}(M \text{ accepts in at most $N$ steps}) \\ &= \lim_n \; \operatorname{Prob}(M \text{ accepts in at most $n$ steps})\,. \end{align*}$$

For all $m$ and $n$, if $m\leq n$ then, for all randomness strings $r$,

$\text{on randomness } r, \: M \text{ accepts in at most } m \text{ steps}$
$\iff$
$(\exists t)\big((\text{on randomness } r, \: M \text{ accepts after exactly } t \text{ steps}) \: \text{ and } \: t\leq m\big)$
$\implies$
$(\exists t)\big((\text{on randomness } r, \: M \text{ accepts after exactly } t \text{ steps}) \: \text{ and } \: t\leq n\big)$
$\iff$
$\text{on randomness } r, \: M \text{ accepts in at most } n \text{ steps}$

.


For all $m$ and $n$, if $\: m\leq n \:$ then $\operatorname{Prob}(\hspace{.02 in}M \text{ accepts in at most } m \text{ steps}) \; \leq \; \operatorname{Prob}(\hspace{.02 in}M \text{ accepts in at most } n \text{ steps}) \;\;\;\;$.



$\frac12 \;\; < \;\; \operatorname{Prob}(\hspace{.02 in}M \text{ accepts})$

$\iff$

$\frac12 \;\; < \;\; \displaystyle\lim_n \: \operatorname{Prob}(\hspace{.02 in}M \text{ accepts in at most } n \text{ steps})$

$\iff$

$(\exists n)\left(\frac12 < \operatorname{Prob}(\hspace{.02 in}M \text{ accepts in at most } n \text{ steps})\hspace{-0.02 in}\right)$