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The Kunita-Watanabe Inequality says:

Let $X,Y$ be two continuous locale martingales and $H,G$ two product-measurable functions on $(0,\infty)\times \Omega$, then $$ \int_0^t|G_s||H_s|d|\langle X,Y \rangle|_s \le \sqrt{\int_0^tG^2_s d\langle X\rangle_s}\sqrt{\int_0^tH^2_s d\langle Y\rangle_s} $$

Suppose I have proved the inequality for the following set of functions:

$$\mathcal{C}:=\{G=\sum_{i=1}^ng_i \mathbf1_{(t_i,t_i+1]}, n\in \mathbb{N}\mbox{ and $g_i$ bounded and measurable}\}$$

Using Monotone Class Theorem I want to extend this first to $G\in \mathcal{C}$ and $H$ bounded and product-measurable. And in a second step to $G,H$ both product-measurable.

In our class we used the following Monotone Class Theorem: Link (Theorem 2).

How do you choose $\mathcal{K}$ and $\mathcal{H}$ in this setting (in both steps)? In addition, product-measurable means with respect to the product $\sigma$-algebra $\mathcal{B}(0,\infty)\otimes \mathcal{F}$. Can the Kunita-Watanabe Inequality be applied to functions, which are measurable with respect to the predictable sigma field on $(0,\infty)\times \Omega$? The predictable sigma field is generated by all adapted and left continuous processes.

Thanks for your help.

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    This isn't the kind of approximation argument necessary for this problem. The trouble is that if we have (G, H) for which this holds and (G, I), say, then it isn't a priori clear to me that this holds for (G, H + I). So we can't show that the set of functions for which this holds (the candidate $\cal{H}$) is a vector space. (A succinct proof of this inequality is given on [George Lowther's blog](https://almostsure.wordpress.com/2010/01/19/properties-of-quadratic-variations/), Theorem 6.)2014-12-17

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