A local martingale is a martingale iff it is in the class DL.

The condition: for every $t\in[0,\infty)$

$$E[\sup\limits_{0\leq s\leq t} |M_s|]<\infty\tag1$$

guarantees a local martingale $M$ is a martingale by ensuring it satisfies the condition for being in the class DL. Moreover, by Burkholder-Davis-Gundy, this means: for every $t\in[0,\infty)$,

$$E[\langle M\rangle^{1/2}_t]<\infty$$

My question is: do there exist (simple?) examples of continuous **martingales** where these two conditions are violated?

What I am really asking is that these are known to be sufficient conditions to guarantee local martingales are martingales, why are they not necessary?