If $g(x,y)$ measurable, why $g$ can be boundedly approximate by functions of the form $\sum_{k=1}^n f_k(x)h_k(y)$?

Question

Let $g=g(x,y)$ measurable.

1) What does mean "$g$ can be boundedly approximate by the sequence $g_n$" ? What is this "boundedly" ?

2) Why $g$ can be boundedly approximate by functions of the form $\sum_{k=1}^n f_k(x)h_k(y)$ ? I didn't find such a result in Real-analysis of Stein and Shakarchi, neither on wikipedia. Is it a classical result ?

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Here the context were I found it : It's in the book Stochastic Differential Equation of Oksendal (See red arrow).

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Answer 1

I don't think Oksendal's approximation assertion is correct. A valid approach would be the use of the functional form of the monotone class theorem. Oksendal wants to assert that two expressions involving a bounded measurable function $g(x,y)$ are equal for all such $g$, and he shows that they are equal if $g$ has the special form $\sum_kf_x(x)h_k(y)$. The function MCT is the perfect tool for such situations. See, for example, https://en.wikipedia.org/wiki/Monotone_class_theorem#Monotone_class_theorem_for_functions

Answer 2

Here is the implementation of John Dawkins' answer, for future reference.

Let $\mathcal{H}$ be the collection of all bounded measurable functions $g(x,\omega)$ that are independent of $\mathcal{F}_t^{(m)}$ and that satisfy: $$\mathbb{E}[g(X_t, \omega) | \mathcal{F}_t^{(m)}] = \mathbb{E}[g(y,\omega)]_{y=X_t}.$$

If $f, g \in \mathcal{H}$ and $c \in \mathbb{R}$, then $f+g$ and $cf$ are in $\mathcal{H}$.

Let $\mathcal{A}$ be the collection of "rectangle" subsets of $\mathbb{R}^n \times \Omega$ of the form $B \times E$, with $B \in \mathcal{B}(\mathbb{R}^n)$ and $E \in \mathcal{F}$. All $\mathbb{1}_{B \times E}$ are in $\mathcal{H}$, and $\mathcal{A}$ is closed under finite intersections.

Finally, suppose $g_n$ is an increasing sequence of functions in $\mathcal{H}$ that converge a.e. to a bounded function $g$. Then we have: \begin{align}\mathbb{E}[g(X_t,\omega)|\mathcal{F}_t^{(m)}] &= \mathbb{E}[\lim_{n \to \infty} g_n(X_t,\omega)|\mathcal{F}_t^{(m)}]\\ &= \lim_{n \to \infty} \mathbb{E}[g_n(X_t,\omega)|\mathcal{F}_t^{(m)}] \quad \text{(by bounded convergence)}\\ &= \lim_{n \to \infty} \mathbb{E}[g_n(y,\omega)]_{y=X_t} \quad\,\, \text{(since $g_n$ are in }\mathcal{H})\\ &= \mathbb{E}[\lim_{n \to \infty} g_n(y,\omega)]_{y=X_t} \quad\,\, \text{(by bounded convergence)}\\ &= \mathbb{E}[g(y,\omega)]_{y=X_t} \end{align}

Therefore $\mathcal{H}$ satisfies the conditions of the Monotone Class Theorem for functions, so $\mathcal{H}$ contains all bounded measurable functions $g(x,\omega)$ that are independent of $\mathcal{F}_t^{(m)}$.

Answer 3

This is can be treated by a monotone class argument.

Let $(X,\mathcal{A})$ and $(Y,\mathcal{B})$ be measurable spaces, and let $(X\times Y,\mathcal{A}\otimes\mathcal{B})$ be the product space ($\mathcal{A}\otimes\mathcal{B}$ is the $\sigma$--algebra generated by the sets $A\times B$, where $A\in\mathcal{A}$ and $B\in\mathcal{B}$)

The space of bounded functions $\mathcal{V}$ that can be the uniformly approximated by finite linear conbinations of functions of the form $\phi(x)\psi(y)$, where $\phi$ and $\psi$ are $\mathcal{A}$ and $\mathcal{B}$ measurable, is a linear space which contains the multiplicative class

$$\mathcal{M}:=\{\sum^n_{j=1}\phi_j(x)\psi_j(y): n\in\mathbb{Z}_+, \phi_j\quad\text{is}\quad\mathcal{A}-\text{measurable},\, \psi_j\quad\text{is} \quad\mathcal{B}-\text{measureable}\}$$

Since $\mathcal{V}$ is closed under taking limits of monotone convergent uniformly bounded sequences, $\mathcal{V}$ contains all the functions that are measurable with respect to the $\sigma$--algebra generated by $\mathcal{M}$.

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A good place to look at different versions of the monotone class theorem is the Klaus Bichteler's Stochastic Integration with Jumps, or Integration: a functional approach. Here is a version that is useful for the OP purposes:

Given an $\Omega$ be an arbitrary nonempty set, $\mathcal{B}_b(\Omega;\mathbb{F})$, where $\mathbb{F}$ is either $\mathbb{R}$ or $\mathbb{C}$, denotes the space of bounded functions on $\Omega$ with values in $\mathbb{F}$.

1. A collection $\mathcal{V}\subset\mathbb{R}^\Omega$, is a monotone class if it is closed under taking pointwise limits of monotone convergent sequences. A collection $\mathcal{V}\subset\mathcal{B}_b(\Omega;\mathbb{R})$ is a bounded monotone class if it is closed under taking pointwise limits of uniformly bounded monotone sequences.

2. A collection $\mathcal{V}\subset\mathcal{B}_b(\Omega;\mathbb{F})$, where $\mathbb{F}=\mathbb{R}$ or $\mathbb{F}=\mathbb{C}$, is a bounded class if it is closed under taking pointwise limits of uniformly bounded convergent sequences.

3. A collection $\mathcal{M}\subset\mathbb{R}^\Omega$ is a real multiplicative class if it is closed under finite multiplication.

Theorem: (Real monotone class theorem) Suppose $\mathcal{V}\subset\mathbb{R}^\Omega$ (resp. $\mathcal{V}\subset\mathcal{B}_b(\Omega;\mathbb{R})$) is a real vector space containing the constant functions, and a monotone (resp. a bounded monotone) class. If $\mathcal{M}\subset\mathcal{V}\cap\mathcal{B}_b(\Omega;\mathbb{R})$ is a multiplicative class, then $\mathcal{V}$ contains all real valued $\sigma(\mathcal{M})$--measurable functions.