  We study a variant of domination, called Roman domination, where we must
assign to each vertex one of the labels 0, 1, or 2 and require that every
vertex with label 0 has a neighbour with label 2. We study the problem of
finding a low-cost Roman dominating function on Lebesgue-measurable graphings,
that is, on infinite graphs whose vertices are the points of a probability
space. We provide a framework to tackle optimisation problems in the measurable
combinatorial setting. In particular, we fully answer the Roman domination
problem on irrational cycle graphs, a specific type of graphing on the space
$\mathbb{R}/\mathbb{Z}$ where an irrational number ${\alpha}$ is given and two
vertices are adjacent if and only if their distance is ${\alpha}$.
