Let $(\Omega, \mathcal{F}, P)$ be a probability space, $X$ an integrable random variable, $\mathcal{G} \subset \mathcal{F}$ a $\sigma$-field. The conditional expectation of $X$ given $\mathcal{G}$ is by definition the unique random variable $Y$ which is $\mathcal{G}$-measurable and satisfies $E[Y;A] = E[X;A]$ for all $A \in \mathcal{G}$. Proving the uniqueness of $Y$ is easy, but existence is harder. I am looking for a nice existence proof with minimal prerequisites.
The traditional proof is to invoke the Radon-Nikodym theorem: the signed measure $\nu(A) = E[X;A]$ on $(\Omega, \mathcal{G})$ is absolutely continuous to $\mu = P|_\mathcal{G}$, so take $Y$ to be the Radon-Nikodym derivative, and it clearly has the desired properties. But the proofs I know of the Radon-Nikodym theorem, while elementary, are somewhat involved (at least 2 pages, even if you only do the absolutely continuous case).
Another proof is to first take $X$ with finite variance, and note that $K = L^2(\Omega, \mathcal{G}, P)$ is a closed subspace of the Hilbert space $H = L^2(\Omega, \mathcal{F}, P)$; then take $Y$ to be the orthogonal projection of $X$ onto $K$. Again, it is then easy to see that $Y$ has the desired properties. But this is not as suitable for students with no functional analysis background. You can develop the necessary facts from scratch but it's a little tedious.
So I am wondering if anyone knows of a simple proof, preferably using only basic measure theory and probability facts.