Meaning of "for each value $X = x$, the random variable Y can be represented in the form $Y = \beta_0 + \beta_1 x + \epsilon$" in linear regression In what follows assume $Y: \ \Omega \to \mathbb R$ and $X: \ \Omega \to E$ 
The following quoute is from page 700 in DeGroot and Schervish - Probability and statistics, introducing a simple linear regression of $Y$ on $X$.
"for each value $X = x$, the random variable Y
can be represented in the form $Y = \beta_0 + \beta_1 x + \epsilon$, where $\epsilon$ is a random variable that has
the normal distribution with mean 0 and variance $\sigma^2$." 

This is understandable in intuitive terms but what precise (probabilistic) object is it referring to in precise mathematical terms? 

Is it referring to $\phi(x)=E[Y|X=x]$, which is such that $\phi(X(\omega))=E[Y|X](\omega)$. But this seems confusing since for fixed $x, \ \phi(x)$ is a single point in $\mathbb R$ and it seems odd that this then should follow some normal distribution (that is take more than one value). 
My take of the assumption is that is means that there exists a regular conditional distribution $\kappa_{Y,\sigma(X)}$ of $Y$ given $\sigma(X)$, such that for any $\omega$ such that $X(\omega)=x$
$$P[\{Y \in B\} |X=x]=\kappa_{Y,\sigma(X)}(\omega,B)=\frac{1}{(2 \pi \sigma ^2)^{n/2 } }\int_B \exp [-\frac{1}{2\sigma^2}\sum_{i=1 }^n(y_i-\beta_0-\beta_1 x_i)^2]$$
But is this correct? Any help would be much appreciated?
 A: Stipulating that $X$ is a random variable is unnecessarily restrictive, so let's consider a generalization. 
I would like to suggest one possible interpretation of this model can be formulated in terms of a (bilinear) function
$$\phi:E\times F\to F;\quad \phi(x,e) = \beta_0+x\beta_1 + e$$
where $E$ and $F$ are understood to be vector spaces over $\mathbb R,$ $\beta_0\in F,$ and $\beta_1$ represents a linear transformation from $E$ to $F.$
Whence, given a random variable $\varepsilon:\Omega\to F$ and given any $x\in E,$ one can form the composition of $\phi$ with the product of the identity map $I_E$ and $\varepsilon$ to obtain
$$Y = \phi \circ \left(I_E \times \varepsilon\right): E\times\Omega\to F;\quad Y(x,\omega) = \beta_0+x\beta_1 + \varepsilon(\omega).$$
This interprets $Y$ as a family of random variables parameterized by $x\in E.$  If you like--because the definition fits--it's a special kind of stochastic process.

When $X:\Omega\to E$ is a random variable, one may also use the diagonal injection
$$\iota:\Omega\to\Omega\times\Omega;\quad \iota(\omega)=(\omega,\omega)$$
(which is always measurable in the category of probability spaces) and consider the composition
$$Y=\phi \circ \left(X \times \varepsilon\right) \circ \iota: \Omega\to F;\quad Y(\omega) = \beta_0+X(\omega)\beta_1 + \varepsilon(\omega),$$
which always is a random variable.  This appears to be the intended meaning of the quoted material in the question.

An obvious variant of this approach will permit you to construct the random variable $(X,Y)$ (taking values in $E\times F$) out of $\phi$ and the bivariate random variable $(X,\varepsilon),$ in terms of which you may speak of conditional distributions.  The point of my original formulation is that conditional distributions are not an essential part of the model, its mathematical formulation, or of the underlying concept.  You can employ them or not depending on the application.
There are many applications where $X$ is not a random variable.  The archetype is a designed experiment, where $X$ represents values of variables that are determined by the experimenter.
