Precise definition of a sample

I am learning mathematical statistics and trying to explain its notions precisely in terms of probability spaces and measurable functions (i.e. random variables). First, I understood what a statistical population is: it can be considered as a probability space $(\Omega,\mathcal{A},\mathbb{P})$ endowed with a random variable $\xi:\Omega\longrightarrow \mathbb{R}$, isn't it? For example, suppose we are investigating the weight of people living in some city. Then the set of people can be regarded as a statistical population at the moment we endow it with a random variable $w$ - the function, which maps each person $P$ to his o her weight $w(P)$. Is it okay?

What I want to understand is what a random sample of length $n$ is. There is a definition in Wikipedia: a random sample of length $n$ is a set of $n$ independent, identically distributed random variables $X_1,X_2,\ldots X_n$. Could you explain why this is a good definition agreed with intuition?

• "subconscious comprehension" would more usually be called "intuition" – Gregory Grant Mar 12 '15 at 19:56