  For a general adapted integrable right-continuous with left limits (RCLL)
process $(X_t)_{t\in[0,\tau]}$ taking values in a metric space $(\mathcal
E,d)$, we show (among other things) that for every $m\in(1,\infty)$
  $$
  \frac{m-1}{2m-1}\|\sup_{t\in[0,\tau]}\mathbb{E}(d(X_{t-},X_\tau)|\mathcal
F_t)\|_m\le \|\sup_{t\in[0,\tau]}d(X_0,X_t)\|_m\le c\frac{m^2}{m-1}
\|\sup_{t\in[0,\tau]}\mathbb{E}(d(X_{t-},X_\tau)|\mathcal F_t)\|_m
  $$
  with a universal constant $c$. This is a probabilistic version of
Fefferman--Stein estimate for the sharp maximal functions. While the former
inequality is derived easily from Doob's martingale inequality, the later
inequality is a consequence of John--Nirenberg inequalities for weighted BMO
processes, which are obtained in this note. We explain how John--Nirenberg
inequalities can be utilized to obtain inequalities for martingales, both old
and new alike in a unified way.
