  The notions of almost everywhere (a.e.) domination and its uniform version
were introduced and studied in reverse mathematics. This paper studies these
notions from a recursion-theoretic point of view and explore their connections
to notions such as randomness and genericity. It is shown that if $Z$ is a.e.
dominating then each $1$-$Z$-random is $2$-random. In other words, $0'\leq_{\rm
LR} Z$ for every a.e. dominating $Z$, where ${\rm LR}$ denotes low-for-random
reducibility. Other results and corollaries are also given.
