  In this survey we discuss work of Levin and V'yugin on collections of
sequences that are non-negligible in the sense that they can be computed by a
probabilistic algorithm with positive probability. More precisely, Levin and
V'yugin introduced an ordering on collections of sequences that are closed
under Turing equivalence. Roughly speaking, given two such collections
$\mathcal{A}$ and $\mathcal{B}$, $\mathcal{A}$ is below $\mathcal{B}$ in this
ordering if $\mathcal{A}\setminus\mathcal{B}$ is negligible. The degree
structure associated with this ordering, the Levin-V'yugin degrees (or
LV-degrees), can be shown to be a Boolean algebra, and in fact a measure
algebra.
  We demonstrate the interactions of this work with recent results in
computability theory and algorithmic randomness: First, we recall the
definition of the Levin-V'yugin algebra and identify connections between its
properties and classical properties from computability theory. In particular,
we apply results on the interactions between notions of randomness and Turing
reducibility to establish new facts about specific LV-degrees, such as the
LV-degree of the collection of 1-generic sequences, that of the collection of
sequences of hyperimmune degree, and those collections corresponding to various
notions of effective randomness. Next, we provide a detailed explanation of a
complex technique developed by V'yugin that allows the construction of
semi-measures into which computability-theoretic properties can be encoded. We
provide two examples of the use of this technique by explicating a result of
V'yugin's about the LV-degree of the collection of Martin-L\"of random
sequences and extending the result to the LV-degree of the collection of
sequences of DNC degree.
