## 2 Generalizations/Modifications of [PERSON]'s Theorem
### Extending the Result to More General Random Variables  
The original theorem, published in 1931, refers to binary random variables. In 1937, [PERSON] himself showed [6] that it also holds for real-valued random variables. This was extended to much more general cases in 1955 by [PERSON] and [PERSON] [7]. They allow random variables that take values from a variety of very general spaces; one of the most general examples is a Borel measurable space (Borel space, for short; see the definition and explanation of related concepts in Appendix A). This space includes all cases that are likely to be encountered in applications. To formally present the generalization of [PERSON]'s Theorem in a form similar to Theorem 1, let \(S\) denote the space from which the random variables take their values, and let \(\mathcal{P}(S)\) be the family of all probability distributions on \(S\). **Theorem 3** (**Hewitt-Savage Theorem**). : _Let \(X_{1},X_{2},\ldots\) be an infinite sequence of \(S\)-valued exchangeable random variables, where \(S\) is a Borel measurable space. Then there exists a probability measure \(\mu\) on \(\mathcal{P}(S)\), such that for every positive integer \(n\) and for any measurable \(A_{1},\ldots,A_{n}\subseteq S\) the following holds:_  
\[\Pr(X_{1}\in A_{1},\ldots,X_{n}\in A_{n})=\int_{\mathcal{P}(S)}\pi(A_{1}) \cdot\ldots\cdot\pi(A_{n})\,d\mu(\pi)\]  
_where \(\pi\in\mathcal{P}(S)\) denotes a random probability distribution from \(\mathcal{P}(S)\), drawn according to \(\mu\). Furthermore, the mixing measure \(\mu\) is uniquely determined._  
Less formally, we can state it this way: an infinite sequence of \(S\)-valued exchangeable random variables is an \(S\)-valued i.i.d.-mix, whenever \(S\) is a Borel measurable space. Note that here \(\mu\) selects a random distribution from \(\mathcal{P}(S)\), which may be a complex object, while in Theorem 1 this random distribution is determined by a single real parameter \(p\in[0,1]\). An interesting related result (which was actually published in the same paper [7]) is called _Hewitt-Savage 0-1 Law_. Let \(X=(X_{1},X_{2},\ldots)\) be an infinite i.i.d. sequence. Further, let \(\mathcal{E}\) be an event that is determined by \(X\). We say that \(\mathcal{E}\) is _symmetric_ (with respect to \(X\)) if the occurrence or non-occurrence of \(\mathcal{E}\) is not influenced by permuting any finite initial segment of \(X\). For example, the event that \"\(X\) falls in a given set \(A\) infinitely many times\" is clearly symmetric, as it is not influenced by permuting any finite initial segment of \(X\). The Hewitt-Savage 0-1 Law says that any such symmetric event has probability either 0 or 1. As an illustration for Theorem 3, consider the following example:  
**Example 2**. : _Assume we put a number of balls into an urn. Each ball has a color, one of \(t\) possible colors (the number of colors may be infinite). Let \(k_{i}\) be the initial number of balls of color \(i\) in the urn, where the \(k_{i}\) are arbitrary fixed non-negative integers. Consider now the following process: draw a ball randomly from the urn, let its color be denoted by \(X_{1}\). Then put back two balls of the same color \(X_{1}\) in the urn. Keep repeating this experiment by always drawing a ball randomly from the urn, and each time putting back two balls of the same color as that of the currently drawn ball. Let \(X_{1},X_{2},X_{3},\ldots\) denote the random sequence of obtained colors. This is called a t-color Polya urn scheme and it is known that the generated sequence \(X_{1},X_{2},X_{3},\ldots\) is exchangeable, see Hill, Lane, and Sudderth [8]. Then, by Theorem 3, the sequence can be represented as an i.i.d.-mix. Note that just from the definition of the urn process this fact may be far from obvious._  
In view of the generality of Theorem 3, one may push further: does the result hold for _completely arbitrary_ random variables? After all, it does not seem self-explanatory why they need to take their values from a Borel measurable space.