# Definition of an unbiased estimator

I have some confusion with the definition of unbiased estimators. Suppose you are given a random sample $$X_1,...,X_n$$ of real-valued random variables defined on a common probability space. A statistic is defined as $$g(X_1,...,X_n)$$, where $$g:\mathbb{R}^n\to\mathbb{R}$$ is a measurable function. If there is some parameter $$\theta$$ of the common distribution of $$X_1,...,X_n$$ that we want to approximate, we might emphasize this with the notation $$g(X_1,...,X_n;\theta)$$, or $$\hat{\Theta}(X_1,...,X_n)$$, in which case, we call this random variable a point-estimator.

Definition: Let $$\varphi:\text{parameters}\to\text{probility measures on} \ \mathbb{R}^n$$, be a parameterization $$\varphi(\theta)=\mu_{\theta}$$. If $$\hat{\Theta}=g(X_1,...,X_n)$$ is an estimator of $$\theta$$, define $$E_{\theta}(\hat{\Theta})=\int_{\mathbb{R}^n}g(x_1,...,x_n)d\mu_{\theta}(x_1,...,x_n)$$. $$\hat{\Theta}$$ is said to be an unbiased estimator if $$E_{\theta}(\hat{\Theta})=\theta$$, for each $$\theta$$.

Question and example: Show that if $$\theta=E(X_1)$$, then $$\hat{\Theta}=\frac{1}{n}\sum\limits_{i=1}^nX_i$$ is an unbiased estimator of $$\theta$$.