Ito's Lemma is proved as Theorem 5 of his paper 'ON A FORMULA CONCERNING STOCHASTIC DIFFERENTIALS'. In his presentation it concerns a function $f$ of $t$ and a $n$-dimensional stochastic process $\xi$ with SDE:
with summation using the Einstein convention. $\beta$ is a $r$-dimensional Brownian Motion.
The only requirements he places on processes $a$ and $b$ is that they be suitably integrable and that they are 'measurable in variables $t$ and $\omega$'. I have not worked through the paper in detail yet and am not familiar with the terminology 'measurable process'. But I am guessing he means 'adapted' to the natural filtration of $\beta$.
I have come across two presentations of Ito's Lemma that claim $a$ and $b$ must be predictable (aka previsible). Unless that's what Ito meant by 'measurable', I can't see that requirement in his paper.
Further, if predictability were a requirement, that would invalidate most uses of the lemma in finance. In particular, the coefficient processes in the geometric Brownian Motion that is assumed by Black-Scholes $$dS_t=S_t\mu dt+S_t\sigma d\beta_t$$ are not predicable, as they are multiples of $S_t$ - even if $\mu$ and $\sigma$ are constants. Nor would the coefficient processes used in most financial applications of the Feynman-Kac Theorem be predictable, as they are often interest rates.
Yet it seems strange that two separate sources would both claim that predictability was a requirement. The sources are this Wikipedia article and Baxter and Rennie's 'Financial Calculus' (see p56 local definition of 'Stochastic Process'). Although both these are just definitions, and one can define anything one wants, they are used in both cases as the requirement for Ito's Lemma to be applicable.
Can anybody advise what K.Ito meant by a 'measurable process', and whether the claims that predictability of coefficients is a requirement is correct?