Typical Martingale Diverges on a Co-$σ$-porous Set

We consider the space of $L^p$-bounded martingales on the Cantor space for $p\in[1,\infty]$. Equipping the Cantor space with two standard metrics, we show that under both, a typical martingale diverges at a typical point. By a 'typical martingale', we mean an element of a co-$σ$-porous set when $p\i