# Question on definition of unbiased estimator

The definition in my textbook is the following.

Let $$\big(\Omega, \mathcal{F}, (P_\vartheta:\vartheta\in\Theta)\big)$$ be a statistical model and $$(E,\mathcal{E})$$ be a measurable space.

If $$E$$ is a finite dimensional vector space, then $$T:\Omega\rightarrow E$$ is called an unbiased estimator for $$\tau:\Theta\rightarrow E$$, if $$\forall \vartheta\in\Theta: E_\vartheta[T]=\tau(\vartheta).$$

Question: Is the expected value $$E_\vartheta[\cdot]$$ always defined on an finite dimensional vector space? It should be a finite dimensional $$\mathbb{R}$$-vector space to be more precise, right?

I can see this if $$T$$ takes discrete values in $$E$$ (for example indexed by the set $$I$$). Then

$$E[T]=\sum_{i\in I} t_i\cdot \underbrace{P(T=t_i)}_{\in\mathbb{R}}\in E$$

But could there be another case I do not see yet? Is the other case just $$(E,\mathcal{E})=(\mathbb{R},\mathcal{B}(\mathbb{R}))$$ as in the definition of wikipedia?