Let $X$ be a local martingale and let $\langle X,X\rangle $ be its associated quadratic variation (or angle bracket) and $\lvert \langle X,X\rangle \rvert$ be the total variation. Given some progressive process, $H$ I want to define $\int_{0}^{T}H_{s} d\langle X,X\rangle _{s}\; (*)$, i.e. the integral wrt the quadratic variation. In all textbooks I have seen that before being able to define $(*)$, we need to make the following assumption:
$$\int_{0}^{T} \lvert H_{s}\rvert \lvert d \langle X, X\rangle\rvert_{s} < \infty \; (**)$$
I do not understand why we $(**)$ is needed because by definition $t \mapsto \langle X,X\rangle _{t}$ is an nondecreasing function (a.s.) such that we can write
$$ \langle X,X\rangle_{t} = \langle X,X\rangle_{t}-0 $$
that is the difference of two nondecreasing functions, and hence of finite variation. Surely the total variation (which is defined as the sum of the two nondecreasing functions) would then simply satisfy
$$\langle X,X\rangle_{t}=\lvert \langle X,X\rangle \rvert_{t} $$
so I think that $(**)$ is redundant, right?
I might be missing something fundamental here, so I am keen to see where I am going wrong, thanks.