Formal book about GLM

I’m searching for a book who treat GLM in a formal way, with a measure theoretic approach. Someone could help me?

---

i try to be more specific

Suppose $ (Omega,mathcal{F},mathbb{P})$ is a probability space, $vec{X} : Omega rightarrow mathbb{R}^n$ random vector and $Y:Omega rightarrow mathbb{R}$ random variable

i can write $Y(omega) = mathbb{E}[Y|vec{X}](omega) + mathcal{E}(omega)$ where $mathbb{E}[Y|vec{X}]$ is the only(a.s) r.variable that satisfy

$$int_{B}mathbb{E}[Y|vec{X}]dmathbb{P}=int_{B}Ydmathbb{P},forall B in sigma(vec{X})$$

by doob-dynkin lemma we can demonstrate that, for some $g: mathbb{R}^n rightarrow mathbb{R}$ measurable function,

$$mathbb{E}[Y|vec{X}](omega)=g(vec{X}(omega))=mathbb{E}[Y|vec{X}=vec{X}(omega)]$$

where $mathbb{E}[Y|vec{X}=vec{X}(omega)]=int_{Omega}Ydmathbb{P}(*|vec{X}=vec{X}(omega))$

ok now my opinion about GLM is to specify the following equations:

• $mathbb{P}(Y in A|vec{X}=vec{x})=int_{A}f_{Y|vec{X}}(y|vec{x}; theta,varphi)dy, forall A in mathcal{B}(mathbb{R})$
• $f_{Y|vec{X}}$ belongs to exponential family
• $mathbb{E}_{vec{beta}}[Y|vec{X}]=g(vec{X}*vec{beta})$

this means that $Y=g(vec{X}*vec{beta})+mathcal{E}$

$forall omega in vec{X}=vec{x}$ holds $Y(omega)=g(vec{x}*vec{beta})+mathcal{E}(omega)$

suppose $vec{x}_1,…,vec{x}_m$ are values of $vec{X}$ that for at least one $omega_i in vec{X}=vec{x}_i$, $y_i=Y(omega_i)$ is observable

Now i always see that $Y_i$ is the $i$– observation, maybe defined as $Y(omega_i)$, and specify the joint distribution saying that these are independent, but.. what? is there a passage to product measure? we can build a set of independent r.v defining on the product space the variable $$Y_i(vec{omega})=Y(omega_i)hspace{1em} mathrm{and} hspace{1em} vec{X}_i (vec{omega})=vec{X}(omega_i)$$

now ${(Y_i,vec{X}_i)}_{i=1,…,m}$ is an independent set of r.v and holds

$$mathbb{P}((Y_1,…,Y_m) in A_1 times…times A_m | vec{X}_1=vec{x}_1,…,vec{X}_m=vec{x}_m)=prod_{i=1}^{m}mathbb{P}(Y_i in A_i |vec{X}_i=vec{x}_i)$$

so the joint conditional density is $$f(vec{y}|vec{x}_1,…,vec{x}_m;vec{theta},varphi)=prod_{i=1}^{m}f_{Y|vec{X}}(y_i|vec{x}_i; theta_i,varphi)$$

this is some use of measure theory and this is what i need to find in a book.