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I am reading a book on stochastic calculus and I am confused about the following:

Martingale representation theorem (some technicalities omitted):

Let $X_t$ be local martingale adapted to Brownian filtration $\mathcal{F}_t$. Then there exists predictable process $H_t$ such that:

$$X_t = X_0 + \int_0^tH_sdW_s$$

After that author gives an example:

Let $X_t = W_t^2 - t$, then $X_t = \int_0^t 2W_sdW_s$ and $H_s = 2W_s$.

But isnt $2W_s$ adapted to $\mathcal{F}_s$? MRT says that $H_s$ should be predictable with respect to $\mathcal{F}_s$

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Since $W_t$ is continuous, being adapted means it is also predictable

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