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The context is that

$X^{(n)}=(X_1,\ldots,X_n)$ consists of $n$ $i.i.d.$ observations according to $F$. Assume $F$ is dominated by a common $\sigma$-finite measure $\mu$, and let $f=\frac{dF(x)}{d\mu}$.

Here, I cannot understand the concept of $f=\frac{dF(x)}{d\mu}$.

How can we differentiate a distribution with respect to a measure?

reference: Romano, Joseph P., Azeem M. Shaikh, and Michael Wolf. "Hypothesis testing in econometrics." Annu. Rev. Econ. 2.1 (2010): 75-104.

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The function $$f(x)=\dfrac{\text dF}{\text d\mu}(x)$$ is the density of the measure $F$ with respect to the dominating measure $\mu$. The notation comes from the unidimensional case when $F$ is the cdf and $\text dμ$ the Lebesgue measure, $\text dx$. It is also called the Radon-Nikodym derivative of the measure $F$ with respect to the measure $\mu$, assuming $F$ is absolutely continuous wrt $\mu$.

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