Recall that two countable groups are measure equivalent, if they admit commuting free actions on the same measure space with finite fundamental domains. The notion of measure equivalence was originally introduced by Gromov as a measure theoretical analog of quasi-isometry. Integral measure equivalence is a refinement of measure equivalence, taking the large scale geometry of the groups into account. We show that if H is a countable group with bounded torsion which is integrable measure equivalence to a right-angled Artin group G with finite outer automorphism group, then H is finitely generated, and H and G are quasi-isometric. This allows us to deduce integrable measure equivalence rigidity results from the relevant quasi-isometric rigidity results for a class of right-angled Artin groups. This is joint work with Camille Horbez.