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Given a random variable $X$ and a sub sigma algebra $N$ of its sampling space, it is often said that $E(\dot \, \mid N)$ is an orthogonal projection, since $X-E(X\mid N)$ and $E( X\mid N)$ are uncorrelated.

I understand that to be $X-E(X\mid N)$ is orthogonal to some subspace of random variables which include $E(X\mid N)$. I wonder how to determine the subspace of random variables from $N$?

Here do we require $X$ to be $L^2$ or $L^1$?

Thanks!

Davide Giraudo
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Tim
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1 Answers1

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In order to make the inner product well-defined, we talk about $L^2(\Omega,\mathcal F,\mu)$, where $(\Omega,\mathcal F,\mu)$ is the underlying probability space. But we then extend condition expectation to integrable random variables.

We use a projection over the closed subspace $L^2(\Omega,\mathcal N,\mu)$, that is, the vector subspace which consists of equivalences classes of $\mathcal N$-measurable random variables (for the relation "equal $\mu$-almost everywhere").

Davide Giraudo
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  • Thanks, Davide! I confused this subspace with the set of random variables which generate the same sigma algebra $N$. could you take a look at it here http://math.stackexchange.com/questions/706924/the-subspace-which-x-ex-mathcal-g-is-orthogonal-to-and-the-set-of-r-v-s-ge – Tim Mar 10 '14 at 17:57