# A question about the definition of sample

Let $$(\Omega,\Sigma,\mu)$$ be a measure space. The definition of a sample is that it is a sequence of iid random variables $$X_1,\ldots,X_n$$. Usually random variables are real valued and the distribution of $$X_i$$ is defined to be $$\mu\circ X_i^{-1}$$

In practice we write things like $$x_1,\ldots,x_n$$ be a sample distributed according to $$\mu$$, where $$x_i$$ are points in the $$\Omega$$. How do we reconcile this with the previous definition? i.e. what are our choices of $$X_i$$ that leads to this situation?

Edit 1: see here for this approach: http://www.math.ntu.edu.tw/~hchen/teaching/StatInference/notes/lecture32.pdf

• Where did you find such definition of sample? – Tim Apr 22 at 19:09