Weighted Average and Expectation in machine learning

Question

Bishop's book defines expectation as "weighted average of a function".

$$E[f(x)] = \int f(x)p(x)\text dx$$

However, the Wikipedia page of weighted function defines a weighted average as

$$E[f(x)] = \frac{\int w(x)f(x)\text dx}{\int w(x)\text dx}$$

Why has Bishop called Expectation as weighted "average" and not just weighted sum as the denominator term is missing in definition of expectation?

Answer 1

When$$\int p(x)\text dx=1$$, $$\mathbb E_p[f(X)]=\dfrac{\int f(x)p(x)\,\text dx}{\int p(x)\,\text dx}$$ The notion is rarely used in probability books, as it does not help (and further depends on the dominating measure chosen as $\text dx$).