"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 $r$,

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

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

<br><br>

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

$\iff$

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

$\iff$

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