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The Radon–Nikodym theorem states that,

given a measurable space (X,Σ), if a σ-finite measure ν on (X,Σ) is absolutely continuous with respect to a σ-finite measure μ on (X,Σ), then there is a measurable function $f$ on X and taking values in [0,∞), such that

$$\nu(A) = \int_A f \, d\mu$$

for any measurable set $A$.

$f$ is called the Radon–Nikodym derivative of $\nu$ wrt $\mu$.

I was wondering in what cases the concept of Radon–Nikodym derivative and the concept of derivative in real analysis can coincide and how?

Thanks and regards!

The Radon–Nikodym theorem states that,

given a measurable space $(X,\Sigma)$, if a $\sigma$-finite measure $\nu$ on $(X,\Sigma)$ is absolutely continuous with respect to a $\sigma$-finite measure $\mu$ on $(X,\Sigma)$, then there is a measurable function $f$ on $X$ and taking values in $[0,\infty)$, such that

$$\nu(A) = \int_A f \, d\mu$$

for any measurable set $A$.

$f$ is called the Radon–Nikodym derivative of $\nu$ wrt $\mu$.

I was wondering in what cases the concept of Radon–Nikodym derivative and the concept of derivative in real analysis can coincide and how?

Thanks and regards!

The Radon–Nikodym theorem states that,

given a measurable space (X,Σ), if a σ-finite measure ν on (X,Σ) is absolutely continuous with respect to a σ-finite measure μ on (X,Σ), then there is a measurable function $f$ on X and taking values in [0,∞), such that

$$\nu(A) = \int_A f \, d\mu$$

for any measurable set $A$.

$f$ is called the Radon–Nikodym derivative of $\nu$ wrt $\mu$.

I was wondering in what cases the concept of Radon–Nikodym derivative and the concept of derivative in real analysis can coincide and how?

Thanks and regards!

The Radon–Nikodym theorem states that,

given a measurable space (X,Σ), if a σ-finite measure ν on (X,Σ) is absolutely continuous with respect to a σ-finite measure μ on (X,Σ), then there is a measurable function $f$ on X and taking values in [0,∞), such that

$$\nu(A) = \int_A f \, d\mu$$

for any measurable set $A$.

$f$ is called the Radon–Nikodym derivative of $\nu$ wrt $\mu$.

I was wondering in what cases the concept of Radon–Nikodym derivative and the concept of derivative in real analysis can coincide and how?

Thanks and regards!