I have seen two (seemingly) different statements called a 'change-of-variables' formula for Lebesgue integration. I'm wondering whether they are equivalent in some sense.
Let $(\Omega, \mathcal{F}, \mathbb{P})$ and $(\mathbb{R}, \mathcal{B}(\mathbb{R}), \mu)$ be two measure spaces. Let $X \colon \Omega \to \mathbb{R}$ and $f \colon \mathbb{R} \to \mathbb{R}$ be measurable functions. Then $$ \int_\Omega f \circ X \, d\mathbb{P} = \int_\mathbb{R} f \, d(\mathbb{P} \circ X^{-1}) $$ where $\mathbb{P} \circ X^{-1}$ is the pushforward measure on $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$. This result is fairly intuitive to me.
Let $\lambda$ be another measure on $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$ such that $\mu$ is absolutely continuous with respect to $\lambda$. Then $$ \int_\mathbb{R} f \, d\mu = \int_\mathbb{R} f \frac{d\mu}{d\lambda} \, d\lambda $$ where $\frac{d\mu}{d\lambda}$ is the Radon-Nikodym density of $\mu$ with respect to $\lambda$. I can sort of see how this result is an extension of the usual Radon-Nikodym theorem (in which $f$ is the identity function).
Can (2) be derived from (1)? Are they equivalent statements of the same 'change-of-variables' result, or different results that have the same name sometimes?