Definition (Consistency)
Let $T_1,T_2,\cdots,T_{n},\cdots$ be a sequence of estimators for the parameter $g(\theta)$ where $T_{n}=T_{n}(X_1,X_2,\cdots,X_{n})$ is a function of $X_{1},X_{2},\cdots,X_{n}.$
The sequence $T_{n}$ is a weakly consistent sequence of estimators for $\theta$ if for every $\varepsilon>0,$ $$\lim_{n\rightarrow\infty}P_{\theta}(|T_{n}-g(\theta)|<\varepsilon)=1.$$ If $T_{n}$ converges with probability one or almost surely (a.s.) to $g(\theta)$, that is, for every $\theta\in\Theta$ $$P_{\theta}\left(\lim_{n\rightarrow\infty}T_{n}=g(\theta)\right)=1,$$ then it is strongly consistent.
Strongly consistency implies weakly consistency.This definition says that as the sample size $n$ increases,the probability that $T_{n}$ is getting closer to $\theta$ is approaching $1$.
I am confused about what is the ${\color{Red}{\left(\Omega,\mathcal{F},P_{\theta}\right)}}$ those $T_{n},\,n=1,2,\ldots$, defined on? What's the specific probability measure ${\color{Red} {P_{\theta}}}$ ?
The parameter $\theta \in \Theta$ usually indicates that the random variables follow a particular family of distributions with parameter $\theta$. In such a probability model, the explicit probability space is usually left unspecified because (a) it is an unnecessary detail for the problem in hand, and (b) the model does not depend on the particular probability space used to represent the model. However, I can still provide a probability space in this case, if you so insist.
Suppose we have a family of cumunative distribution functions $ F_\theta: \mathbb{R} \to [0, 1] $. For each $\theta \in \Theta$, the Carathéodory construction gives a Borel probability measure $ Q_\theta: \mathcal{B}(\mathbb{R}) \to [0, 1] $, called the Lebesgue-Stieltjes measure, that satisfies $ F_\theta(x) = Q_\theta((-\infty, x]) $ for all $x$. Next, consider the countably infinite product probability space
$$ (\Omega, \mathcal{F}, P_\theta) := (\mathbb{R}^\mathbb{N}, \mathcal{\mathcal{B}(\mathbb{R})}^{\otimes \mathbb{N}}, Q_\theta^{\otimes \mathbb{N}}) $$
If you don't know about the Lebesgue-Stieltjes measure or infinite product probability, you can consult any measure-theoretical probability textbook.
Anyway, we now define the random variables $X_n, n \in \mathbb{N}$ on $(\Omega, \mathcal{F}, P_\theta)$ simply to be the coordinate projections, i.e. $X_n(\omega) = \omega_n$ for $\omega \in \mathbb{R}^\mathbb{N}$. Then $X_n, n \in \mathbb{N}$ are i.i.d. with distribution function $F_\theta$.
Next, define $T_n = t_n(X_1,\cdots,X_n)$. This definition just spells out as $T_n(\omega) = t_n(\omega_1,\cdots,\omega_n)$ for $\omega \in \mathbb{R}^\mathbb{N}$. For each fixed $\theta \in \Theta$, the parameter $g(\theta)$ is simply a number, so it make sense to write
$$ \lim_{n \to \infty} P_{\theta}(|T_{n}-g(\theta)|<\varepsilon) = 1 $$
Spelled out, it means
$$ \lim_{n \to \infty} P_{\theta}\left( \{ \omega \in \mathbb{R}^\mathbb{N} : |t_n(\omega_1,\cdots,\omega_n)-g(\theta)|<\varepsilon \} \right) = 1 $$