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We prove that the volume measure of the Brownian sphere is equal to aconstant multiple of the Hausdorff measure associated with the gauge function$h(r)=r^4\log\log(1/r)$. This shows in particular that the volume measure ofthe Brownian sphere is determined by its metric structure. As a key ingredientof our proofs, we derive precise estimates on moments of the volume of balls inthe Brownian sphere.

The volume measure of the Brownian sphere is a Hausdorff measure