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?


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