Solved – Why doesn’t the exponential family include all distributions

I am reading the book:

Bishop, Pattern Recognition and Machine Learning (2006)

which defines the exponential family as distributions of the form (Eq. 2.194):
p(mathbf x|boldsymbol eta) = h(mathbf x) g(boldsymbol eta) exp {boldsymbol eta^mathrm T mathbf u(mathbf x)}

But I see no restrictions placed on $h(mathbf x)$ or $mathbf u(mathbf x)$. Doesn't this mean that any distribution can be put in this form, by appropriate choice of $h(mathbf x)$ and $mathbf u(mathbf x)$ (in fact only one of them has to be chosen properly!)? So how come the exponential family does not include all probability distributions? What am I missing?

Finally, a more particular question that I am interested in is this: Is the Bernoulli distribution in the exponential family? Wikipedia claims it is, but since I am obviously confused about something here, I would like to see why.

First, note there is a terminology problem in your title: the exponential family seems to imply one exponential family. You should say a exponential family, there are many exponential families.

Well, one consequence of your definition: $$p(mathbf x|boldsymbol eta) = h(mathbf x) g(boldsymbol eta) exp {boldsymbol eta^mathrm T mathbf u(mathbf x)}$$ is that the support of the distribution family indexed by parameter $eta$ do not depend on $eta$. (The support of a probability distribution is the (closure of) the least set with probability one, or in other words, where the distribution lives.) So it is enough to give a counterexample of a distribution family with support depending on the parameter, the most easy example is the following family of uniform distributions: $ text{U}(0, eta), quad eta > 0$. (the other answer by @Chaconne gives a more sophisticated counterexample).

Another, unrelated reason that not all distributions are exponential family, is that an exponential family distribution always have an existing moment generating function. Not all distributions have a mgf.

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