My analysis professor told us to take the following theorem for granted in order to prove other results, but I would like to see a proof of it, since I think it will be beneficial. Here is the theorem:
$\textbf{Theorem.}$(Kolmogorov's Existence Theorem)
If probability measure $\mu_k$ on $(\mathbb{R}^k, \mathcal{B}_{\mathbb{R}^k})$, $k\geq 1$, satisfy $$\mu_{k-1}=\mu_k\psi^{-1}_k, \quad k>1, $$ then there exists on $(\mathbb{R}^\infty, \mathcal{B}_{\mathbb{R}^\infty})$ a unique probability measure $P$ satisfying $$\mu_k=P\pi^{-1}_k, \quad k\geq 1. $$
I want to point out that, given $k\in \mathbb{N}$, $\pi_k:\mathbb{R}^\infty \mapsto \mathbb{R}^k$ is the natural projection defined by $$\pi_k(x_1, \ldots , x_k, x_{k+1}, \ldots ) =(x_1,\ldots , x_k).$$
Is there a probability book where I can find a proof of this theorem? I have been searching online! Thank you!
You can find a proof in Section 10 of these lecture notes that I wrote. I tried to make explicit the role played by the topology, particularly compactness, and the connection with Tychonoff's theorem.
It is also in Durrett's Probability: Theory and Examples. I believe also in Folland's Real Analysis and Resnick's Probability Path. Basically any graduate-level probability text should have it.