the representation
$$
\xi=F\left(\xi_{1}, \ldots, \xi_{m}\right)
$$
with $F$ as linear combination of the different elements
$$
e^{i} \sum_{k=1}^{m} \lambda_{k} \xi_{k}, \quad \xi_{k}=\mu\left(\Delta_{k}\right), \quad k=1, \ldots, m
$$
Accordingly, we can see that for all these linear combinations the corresponding is given by the limit
$$
\mathcal{D} \xi=\lim {n \rightarrow \infty} \sum \frac{1}{M(\Delta)} E\left(\xi \mu(\Delta) \mid \mathfrak{A}{\Delta \mid}\right) 1_{\Delta}
$$
taken in $L_{2}(\Theta \times \mathbb{T} \times \Omega)$. The sum is here taken on the sets $\Delta$ of the $n^{\text {th }}$-series of the partitions of $\Theta \times \mathbb{T}$. . Clearly, this limit defines the linear operator $\mathcal{D}$ :
$$
\operatorname{dom} \mathcal{D} \ni \xi \Longrightarrow \mathcal{D} \xi \in L_{2}(\Theta \times \mathbb{T} \times \Omega)
$$
MATH4512 COURSE NOTES ：
Let $I^{P} \varphi_{p}, p=1,2, \ldots$, be the Itô $p$-multiple integrals with respect to a general stochastic measure $\mu$ in the scheme . The anticipating derivative is
$$
\mathcal{D} I^{p} \varphi_{p}=I^{p-1} \hat{\varphi}{p}(\cdot, \theta, t), \quad(\theta, t) \in \theta \times \mathbb{T} $$ with the integrands $$ \hat{\varphi}{p}:=\sum_{j=1}^{p} \varphi_{p}(\ldots, \theta, t, \ldots)
$$
depending on $(\theta, t) \in \Theta \times \mathbb{T}$ as parameter. The couple $(\theta, t)$ comes in at the place of the corresponding couples $\left(\theta_{j}, t_{j}\right), j=1, \ldots, p$. Here we have
$$
|\mathcal{D} \xi|_{L_{2}}=p^{1 / 2}|\xi|, \quad p=1,2, \ldots .
$$
All the elements
$$
\xi=\sum_{p=0}^{\infty} \oplus \xi_{p}: \quad \xi_{0}=E \xi_{p}, \xi_{p}=I^{p} \varphi_{p}, p=0,1, \ldots
$$
with
$$
\sum_{p=1}^{\infty} p\left|\xi_{p}\right|^{2}<\infty
$$