I see this page [Ordinary least square and random projection](https://mathoverflow.net/questions/158270/ordinary-least-square-and-random-projection), and I am thinking that how $L^2$ integrable random variables be regarded as projections over a defined filtration sequence $\mathcal{F\_n}$ ?


It seems that a vector space of square integrable random variables defined on the same probability space is not enough? 
Is there existing literature addressing a formalism of this projection view? Thank you!



Square integrable conditional expectations as projections
No. Vector space structure is not enough, we actually need a compatible lattice structure to make things work. To apply the conditional expectation operator $E(\bullet\mid Y)$ onto the Hilbert space consisting of $L^2(\mathcal{X})$ random variables defined on the probability space $\mathcal{X}$, it requires us to consider following construction. A probability Hilbert space $H$(of $L^2(\mathcal{X})$ with $<x,y>=E(x-Ex)(y-Ey)$) if it satisfies following conditions



> 
> (1) it has unitary element $\boldsymbol{1}\_{\mathcal{X}}$.
> 
> 
> (2) it adopts a lattice structure compatible with the vector space
>  structure $(H,\ge)$ such that $x\vee\boldsymbol{1}=\boldsymbol{0}$ iff
>  $x=\boldsymbol{0}$
> 
> 
> (3) For $x\ge \boldsymbol{0}$ and $y\ge \boldsymbol{0}$ $<x,y>\geq 0$.
>  equality holds iff $x\wedge y=\boldsymbol{0}$
> 
> 
> 


A closed subspace $G$ of a probability Hilbert subspace $H$ is a closed probability space such that it contains $\boldsymbol{1}\_G$ and closed under $\vee \wedge$ operations of the lattice. The probability span of $y\in H$ is defined to be the intersection of all probability Hilbert subspaces that contains ${y}$(which is like the way we define ideals/spectrum). Then $E(\bullet\mid y)$ is a projection operator onto the probability span of $y$.


[Small&McLeish] contains more details of construction of probability spans.


[Small&McLeish]Small, Christopher G., and Don L. McLeish. Hilbert space methods in probability and statistical inference. Vol. 920. John Wiley & Sons, 2011.


