I work in this literature. Daily returns are relatively easy to predict. But just predictability is not enough to publish and have an impact. And it is difficult to come up with a fundamentally important story for highfrequency return predictability.
Why most empirical asset pricing papers do not predict daily returns?

Yes. Suppose there is some true underlying price p*_t, which is a martingale, and you observe p_t = p*_t + eps_t. Now consider the observed return over j periods,
r_{t,t+j} = p_{t+j}  p_t = r*_{t,t+j} + eps_{t+j}  eps_t
where r*_{t,t+j} = p*_{t+j}  p*_t is the true return.
the measured return is thus equal to the true return plus noise with variance equal to 2*var(eps). The variance of the signal, r*_{t,t+j}  i.e., the thing you really want to measure  is going to be proportional to j. Errors in the prices you observe will therefore contribute a constant amount of variance to the observed returns, while the signal component, r*, will contribute a variance that grows with the horizon.
So if you use monthly returns, the signaltonoise ratio is 21 times higher than for daily returns.I love how you write that "p*_t is a martingale" but don't define your filtration properly. So is a "monthly" participant just going to observe a sequence of month end prices, and ignore the daily data? Clearly, the filtration generated by {day 0, day 1, ..., day 30} is bigger than the filtration generated by just {day 30}. A martingale with respect to the latter, is not a martingale with respect to the former.
Ignoring all that stochastic analysis stuff, your measurement error assumption makes no sense. Are you literally telling me  of all things  prices of publicly traded stocks are recorded with error? Especially at the daily level?
Finally, before screaming on how finance people are dumdums, perhaps you should think closer on the stochastic properties of time series data. There are many reasons why finance academia prefers monthly over daily data, but "measurement error" is definitely not one of them.JFC, you learned a little stochastic calculus and think you're a genius. Spare us.
The far bigger issue than the filtration (which is implied by context) is the fact that there is a massive assumption of (1) one noise DGP and (2) those noise components being uncorrelated. The first assumption is garbage because there is the noise of information diffusion, the error induced by rounding prices, and other sources of noise (e.g. option arbitrage) which have a different structure. That leads to the second assumption being trash: it's not only unlikely, it disagrees with the data. We should expect option arbitrageinduced noise terms to be negatively correlated with the other noise terms; and, the rounding error noise might normally be small but in some cases it can be problematic.
Furthermore, you really have a marked process where the marking (size) is correlated with the magnitude of these noise terms. I mean, you didn't even try for something that sophisticated; but, if you've read any microstructure literature, you should have. But hey, you keep up your onanistic comments about filtrations when there are far more important issues that make forecasting daily returns more difficult. (Not impossible, just more difficult.)

I work in this literature. Daily returns are relatively easy to predict. But just predictability is not enough to publish and have an impact. And it is difficult to come up with a fundamentally important story for highfrequency return predictability.
Let me correct you right there—daily MARKET returns are relatively easy to predict. Cross section prediction in the daily frequency on the other hand is tough enough that few bother to try it.

what is a good predictor for daily market returns?
I work in this literature. Daily returns are relatively easy to predict. But just predictability is not enough to publish and have an impact. And it is difficult to come up with a fundamentally important story for highfrequency return predictability.
Let me correct you right there—daily MARKET returns are relatively easy to predict. Cross section prediction in the daily frequency on the other hand is tough enough that few bother to try it.