  In metric Diophantine approximation, one frequently encounters the problem of
showing that a limsup set has positive or full measure. Often it is a set of
points in $m$-dimensional Euclidean space, or a set of $n$-by-$m$ systems of
linear forms, satisfying some approximation condition infinitely often. The
main results of this paper are bootstraps: if one can establish positive
measure for such a limsup set in $m$-dimensional Euclidean space, then one can
establish positive or full measure for an associated limsup set in the setting
of $n$-by-$m$ systems of linear forms. Consequently, a class of $m$-dimensional
results in Diophantine approximation can be bootstrapped to corresponding
$n$-by-$m$-dimensional results. This leads to short proofs of existing, new,
and hypothetical theorems for limsup sets that arise in the theory of systems
of linear forms. We present several of these.
