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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

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

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