Let　H∞δ　be　the　Hausdorff　content　on　Rn　of　order　δ∈(0,n].　For　the　space　BMO(H∞δ)　of　bounded　mean　oscillation　that　is　defined　with　respect　to　H∞δ,　the　authors　establish　its　equivalent　characterizations　via　replacing　the　L1(H∞δ)-integrability　in　its　definition　by　the　Luxemburg　type　Lφ(H∞δ)-integrability,　where　φ　can　be　either　a　convex　or　a　concave　nonnegative　function.　Typical　examples　of　φ　include　φ(t)=tp,　φ(t)=ept−1　and　φ(t)=log+⁡∘⋯∘log+⁡t,　where　p∈(0,∞)　and　log+⁡t:=max⁡{log⁡t,0}.　©　2025　Elsevier　Inc.