I need help to show this equality.
$(*)$ $\{\left<X\right> _ {\infty} < \infty \} = \{ [X]_\infty < \infty \}$ if ${E}\sup_{n\in T}(\Delta X_n)^2 < \infty$
I've already show $\{\left<X\right>_\beta<\infty\} \subset \{[X]_\beta<\infty\}$ almost surely
I know, because of the inequality from lenglart$${P}(\left[X\right]_\beta\ge k,\left<X\right>_\beta\le m)\le\frac{m}{k}$$ for $k\longrightarrow\infty$ follow, that $$\left\{\left[X\right]_\beta=\infty\right\}=\bigcap_{k=1}^{\infty}{\left\{\left[X\right]_\beta\ge k\right\}}.$$ This imply this equation $${P}(\left[X\right]_\beta=\infty, \left<X\right>_\beta\le m)=0.$$
For the predicable quadratic variation aplies $$\left\{\left<X\right>_\beta<\infty\right\}=\bigcup_{m=1}^{\infty}{\left\{\left<X\right>_\beta\le m\right\}}.$$ for $m\longrightarrow\infty$ follows $${P}([X]_\beta=\infty,\left<X\right>_\beta<\infty)=0.$$ The consequence is the claim $$\left\{\left<X\right>_\beta<\infty\right\}\subset\left\{\left[X\right]_\beta<\infty\right\} \textrm{almost surely}.$$
But now I don't know, how to use this property ${E}\sup_{n\in T}(\Delta X_n)^2 < \infty$ to show $(*)$.
I hope, you understand, what I mean. My english is not the yellow from the egg.