Linear and almost linear functions on a measurable Hilbert space

Let $X$ be a separable Hilbert space, $\mathfrak B$ be the Borel $\sigma$-algebra in $X$, and ($\mu$ be a probability measure on $\mathfrak B$. A function $\varphi(x)$ is called a $\mu$-measurable linear function if it is the limit in $\mu$ of a sequence of continuous linear functions. A function $\varphi(x)$ is called an almost linear function, if it is $\mathfrak B$-measurable and there exists a linear $\mathfrak B$-measurable manifold $L\subset X$ such that $\mu(L)=1$ and $\varphi(x)$ is linear on $L$.
We investigate the class of all linear functions and (in the case of quasiinvariant measure) the class of all almost linear functions.