The volume measure of the Brownian sphere is a Hausdorff measure

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