Denote the Lower Box dimension as $\underline\dim_B$ and the packing dimension as $\dim_P$, is there a criterion to tell, for a given set $F$, when $\underline\dim_B(F)<\dim_P(F)$? Or, at least, are there examples for both sides of the inequality?

Update: note that $h\cdot M$ is a martingale and then the first moment vanishes, result follows.

But why is the middler term vanishing?

The Itô's isometry yields

To show this equality holds, we first square both sides and then take expectations on both sides, then we have

We wish to show that $\mathcal{H}$ is a complete subspace of $(\Omega,\mathcal{F}_\infty,\mathbb{P})$ and then by the Hilbert space structure it is closed. To show that we need the equaltiy