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Please could you help me find a rigorous mathematical definition of sampling as it is used in mathematical statistics?

Let ##X:\Omega\rightarrow\mathbb{R}## be a random variable and ##X_{1},...,X_{n}## is a statistical sample. What is its mathematical meaning in probability theory? Are we talking about elements of the sample space ##\omega_{1},...,\omega_{n}## from ##\varOmega^{n}## ?

We always say that samples are equally probable and independent. But in order for ##\omega_{i}## to be equally probable we need to have on ##\varOmega## a uniform discrete distribution that assigned equal probability to every element ##\omega## . But this means that ##\varOmega## must be finite and the probability measure defined on it must be uniform. But what if the original measure on ##\varOmega## was not uniform and it assigned different probabilities to different elements? Then we cannot have samples that are equally probable.

Let me reiterate - a discrete uniform distribution cannot be defined on an infinite set, therefore if we sample such a set we will not be able to say that all samples we might draw are equally probable. This is why for instance we cannot have a notion of a completely random real or even integer number. On a set of reals we cannot define a distribution that assigns equal probability to individual points. If we were to draw samples which would not be equally probable and mostly independent, then what would be the point? All of the basic formulas of statistics would not apply. Take for instance the definition of a sample mean or variance and see if it works on samples which don't have equal probability or samples of zero probability. What would the mathematical meaning of this sampling be?

Does it mean that the scope of definition of sampling is restricted to finite ##\varOmega## with uniform distribution? Does it mean that sampling does not make sense in other scenarios?

I don't suppose this is the case, sampling must mean something in those cases.

Please could you help me find a rigorous definition and recommend me some literature?

Let ##X:\Omega\rightarrow\mathbb{R}## be a random variable and ##X_{1},...,X_{n}## is a statistical sample. What is its mathematical meaning in probability theory? Are we talking about elements of the sample space ##\omega_{1},...,\omega_{n}## from ##\varOmega^{n}## ?

We always say that samples are equally probable and independent. But in order for ##\omega_{i}## to be equally probable we need to have on ##\varOmega## a uniform discrete distribution that assigned equal probability to every element ##\omega## . But this means that ##\varOmega## must be finite and the probability measure defined on it must be uniform. But what if the original measure on ##\varOmega## was not uniform and it assigned different probabilities to different elements? Then we cannot have samples that are equally probable.

Let me reiterate - a discrete uniform distribution cannot be defined on an infinite set, therefore if we sample such a set we will not be able to say that all samples we might draw are equally probable. This is why for instance we cannot have a notion of a completely random real or even integer number. On a set of reals we cannot define a distribution that assigns equal probability to individual points. If we were to draw samples which would not be equally probable and mostly independent, then what would be the point? All of the basic formulas of statistics would not apply. Take for instance the definition of a sample mean or variance and see if it works on samples which don't have equal probability or samples of zero probability. What would the mathematical meaning of this sampling be?

Does it mean that the scope of definition of sampling is restricted to finite ##\varOmega## with uniform distribution? Does it mean that sampling does not make sense in other scenarios?

I don't suppose this is the case, sampling must mean something in those cases.

Please could you help me find a rigorous definition and recommend me some literature?

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