This is given as a counterexample that not all martingales $Y$ with $Y_0 = 0$ can be represented as $H\bullet X$ (= "discrete stochastic integral" ... wherever this term comes from??) for a given martingale $X$ with $X_0 = 0$:
Why does the author write "there is no number"? Shouldn't it be "there is no random variable, such ..." or "there are no numbers, such ..."?
The random variable $H_1$ has to be time $0$ measureable which makes it a constant.