  We investigate the construction of a Hausdorff uo-Lebesgue topology on a
vector lattice from a Hausdorff (o)-Lebesgue topology on an order dense ideal,
and what the properties of the topologies thus obtained are. When the vector
lattice has an order dense ideal with a separating order continuous dual, it is
always possible to supply it with such a topology in this fashion, and the
restriction of this topology to a regular sublattice is then also a Hausdorff
uo-Lebesgue topology. A regular vector sublattice of
$\mathrm{L}_0(X,\Sigma,\mu)$ for a semi-finite measure $\mu$ falls into this
category, and the convergence of nets in its Hausdorff uo-Lebesgue topology is
then the convergence in measure on subsets of finite measure. When a vector
lattice not only has an order dense ideal with a separating order continuous
dual, but also has the countable sup property, we show that every net in a
regular vector sublattice that converges in its Hausdorff uo-Lebesgue topology
always contains a sequence that is uo-convergent to the same limit. This
enables us to give satisfactory answers to various topological questions about
uo-convergence in this context.
