Let me motivate my question with this example.

The volume integral of a ball $\int_{B(0,R)} dx$ can be written as an integral over the surface of balls, i.e.

$$\int_{B(0,R)} dx = \int_0^R \int_{\partial B(0,r)}dS dr.$$

This shows, that the derivative w.r.t. $R$ is just the surface-integral

$$\frac{d}{dR} \int_{B(0,R)} dx = \int_{\partial B(0,R)}dS$$

Now, what happens if we generalize this: Let $F: \mathbb{R}^n \rightarrow \mathbb{R}$ and suppose that $\lambda( F^{-1}(-\infty, R]) < \infty$ for all $R$ then I see two ways to generalize my example:

(1) $\mu((-\infty,R]):=\int_{F^{-1}(-\infty, R]} dx$ defines a measure. The measure $\mu$ is a.c. with respect to the Lebesgue measure, i.e. $\mu(A)= \int_{A} fdx$ for some measurable $f$. In my example, the function $f(r)= \int_{\partial B(0,r)} dS.$

(2) But the most obvious generalization would be probably: $$\int_{F^{-1}(-\infty, R]} dx = \int_{-\infty}^{R} \int_{F^{-1}(\{r\})} dSdr?$$

Apparently, (1) is more general than (2). I suspect (although do not know a proof of this) that (2) holds for submersions. Now my question is: How big is the difference between (1) and (2), i.e. does (1) only hold, if the representation in (2) holds almost everywhere? Under what precise conditions does (2) hold?

Let me motivate my question with this example.

The volume integral of a ball $\int_{B(0,R)} dx$ can be written as an integral over the surface of balls, i.e.

$$\int_{B(0,R)} dx = \int_0^R \int_{\partial B(0,r)}dS dr.$$

This shows, that the derivative w.r.t. $R$ is just the surface-integral

$$\frac{d}{dR} \int_{B(0,R)} dx = \int_{\partial B(0,R)}dS$$

Now, what happens if we generalize this: Let $F: \mathbb{R}^n \rightarrow \mathbb{R}$ and suppose that $\lambda( F^{-1}(-\infty, R]) < \infty$ for all $R$ then I see two ways to generalize my example:

(1) $\mu((-\infty,R]):=\int_{F^{-1}(-\infty, R]} dx$ defines a measure. The measure $\mu$ is a.c. with respect to the Lebesgue measure, i.e. $\mu(A)= \int_{A} fdx$ for some measurable $f$. In my example, the function $f(r)= \int_{\partial B(0,r)} dS.$

(2) But the most obvious generalization would be probably: $$\int_{F^{-1}(-\infty, R]} dx = \int_{-\infty}^{R} \int_{F^{-1}(\{r\})} \frac{1}{||\nabla F(x)||}dS(x) dr?$$

Apparently, (1) is more general than (2). I suspect (although do not know a proof of this) that (2) holds for submersions. Now my question is: How big is the difference between (1) and (2), i.e. does (1) only hold, if the representation in (2) holds almost everywhere? Under what precise conditions does (2) hold?