Decomposition of $L^2$-spaces and singular measures If $\langle \Omega, \mathfrak{F}, \mathbb{P}\rangle$ is a measure space and $L^2$ is the corresponding $L^2$ space and 
$$
K\oplus K^{\perp} \cong L^2(\mathfrak{F},\mathbb{P}).  
$$
Then let:
$$
\mathfrak{F}_1\triangleq \sigma(\{H \in K \}) \\
\mathfrak{F}_2\triangleq \sigma(\{H \in K^{\perp} \}) \\
$$
Moreover, do there exist singular measures $\mu$ and $\nu$ such that
$$
\mu + \nu = \mathbb{P},\\
L^2(\mathfrak{F}_1,\mu) \oplus L^2(\mathfrak{F}_2,\nu) \cong L^2(\mathfrak{F},\mathbb{P})
$$
$$
\int f d\nu =0 = \int g d\mu 
$$
if $f \in L^2(\mathfrak{F}_1)$ and $g \in \mathfrak{F}_2$?
I was thinking of using conditional measures, but I'm not sure how?
 A: I'm not sure I have completely parsed your question, but it seems to be based on an assumption that the $\sigma$-fields $\mathfrak{F}_1, \mathfrak{F}_2$ are in some sense "orthogonal".  That doesn't have to be true; they can even be equal.
Take as an example $\Omega = [0,1]$ with its Borel $\sigma$-field $\mathcal{F}$ and let $\mathbb{P}$ be Lebesgue measure.  Note the following fact: if $g : [0,1] \to \mathbb{R}$ is Borel and injective with a Borel left inverse $g^{-1} : \mathbb{R} \to [0,1]$, then $\sigma(g)$, the smallest $\sigma$-field making $g$ measurable, is $\mathfrak{F}$ itself.  (Proof: for any Borel set $B \subset [0,1]$, the function $1_B \circ g^{-1} \circ g = 1_B$ is $\sigma(g)$-measurable.)
Let $g_1(x) = x$ and $g_2(x) = x^2 - \frac{1}{2}$.  They are orthogonal in $L^2$, so if we let $K$ be the one-dimensional space spanned by $g_1$, we have $g_2 \in K^\perp$.  But each is Borel and injective (indeed, each is a homeomorphism onto its image) so we end up with $\mathfrak{F}_1 = \mathfrak{F}_2 = \mathfrak{F}$.
In particular, if $\nu$ is a measure absolutely continuous to $\mathbb{P}$, and $\int f\,d\nu = 0$ for all $f \in L^2(\mathfrak{F}_1)$, then $\nu = 0$.
Basically, the issue is that there are a lot of operations that don't enlarge a generated $\sigma$-field, but do enlarge a subspace of $L^2$.  Multiplication is perhaps the most obvious example.
