Typically, one starts with a (Lebesgue/Borel measurable) set $B\subset \Bbb{R}^3$, and considers a function $\rho:\Bbb{R}^3\to[0,\infty)$ such that $\rho|_{B^c}=0$, i.e $\rho(x)=0$ for all $x\notin B$. This is what we refer to as the "density of $B$", so of course the interpretation is that for each $x\in\Bbb{R}^3$, $\rho(x)\geq 0$ is a number telling us the density of the body $B$ at the point $x$ (this is why if $x\notin B$, we require $\rho(x)=0$...because if there's is no body there, we should assign the density to be zero). From this density, we can construct a measure $m$ as follows: for any $A\subset \Bbb{R}^3$, we define $m(A):=\int_A \rho\, dV$ (where $dV$ stands for integration with respect to volume, or more precisely, the Lebesgue measure on $\Bbb{R}^3$). This is often notationally denoted as $dm=\rho \,dV$; again the $d$ has no independent meaning in this context. It's sole purpose is to invoke a sense of familiarity with classical notation. The meaning of the equation $dm=\rho\,dV$ is that for all (measurable) sets $A$, $m(A)=\int_A\rho\,dV$ (i.e the measure $m$ is defined by integrating $\rho$ with respect to Lebesgue measure).
Typically, one starts with a (Lebesgue/Borel measurable) set $B\subset \Bbb{R}^3$, and considers a function $\rho:\Bbb{R}^3\to[0,\infty)$ such that $\rho|_{B^c}=0$, i.e $\rho(x)=0$ for all $x\notin B$. This is what we refer to as the "density of $B$", so of course the interpretation is that for each $x\in\Bbb{R}^3$, $\rho(x)\geq 0$ is a number telling us the density of the body $B$ at the point $x$ (this is why if $x\notin B$, we require $\rho(x)=0$...because if there's is no body there, we should assign the density to be zero). From this density, we can construct a measure $m$ as follows: for any $A\subset \Bbb{R}^3$, we define $m(A):=\int_A \rho\, dV$ (where $dV$ stands for integration with respect to volume, or more precisely, the Lebesgue measure on $\Bbb{R}^3$).
Typically, one starts with a (Lebesgue/Borel measurable) set $B\subset \Bbb{R}^3$, and considers a function $\rho:\Bbb{R}^3\to[0,\infty)$ such that $\rho|_{B^c}=0$, i.e $\rho(x)=0$ for all $x\notin B$. This is what we refer to as the "density of $B$", so of course the interpretation is that for each $x\in\Bbb{R}^3$, $\rho(x)\geq 0$ is a number telling us the density of the body $B$ at the point $x$ (this is why if $x\notin B$, we require $\rho(x)=0$...because if there's is no body there, we should assign the density to be zero). From this density, we can construct a measure $m$ as follows: for any $A\subset \Bbb{R}^3$, we define $m(A):=\int_A \rho\, dV$ (where $dV$ stands for integration with respect to volume, or more precisely, the Lebesgue measure on $\Bbb{R}^3$). This is often notationally denoted as $dm=\rho \,dV$; again the $d$ has no independent meaning in this context. It's sole purpose is to invoke a sense of familiarity with classical notation. The meaning of the equation $dm=\rho\,dV$ is that for all (measurable) sets $A$, $m(A)=\int_A\rho\,dV$ (i.e the measure $m$ is defined by integrating $\rho$ with respect to Lebesgue measure).
A measure space $(X,\mathcal{M},\mu)$. Here, $X$ is a set, $\mathcal{M}$ is a collection of "nice subsets" of $X$, and $\mu$ is what's called a measure (which is a function $\mathcal{M}\to [0,\infty]$). Very roughly, if $A\in\mathcal{M}$ (recall this means $A$ is a "nice" subset of $X$) then $\mu(A)$ is a number called the measure of $A$, and intuitively you should think of it$\mu$ as a "measurement device" telling you "how big" the set $A$ is.
A certain collection $\mathcal{L}^1(\mu)$ of "nice" functions $X\to\Bbb{R}$ (we can consider more general target spaces, but let's not bother now), called the Lebesgue-integrable functions on $X$, with respect to $\mu$. Most of the functions you encounter in an introductory physics class are going to be "nice enough" anyway, so I won't bother defining this space now.
A linear mapping $\mathcal{L}^1(\mu)\to\Bbb{R}$ which assigns to each "nice function" $f\in \mathcal{L}^1(\mu)$ a specific real number, denoted as $\int_Xf\, d\mu$ or $\int_Xf(x)\, d\mu_x$$\int_Xf(x)\, d\mu(x)$, called the Lebesgue integral of $f$ over $X$ with respect to the measure $\mu$ (again, I won't go into details of how this is carefully defined unless you really want me to). The $d$ here is just to make things look as classical as possible, don't read too much into this here.
Now given a body $B$, we can regardinterpret its moment of inertia $I=\int_Br^2\, dm$ using the Lebesgue integral as follows. When I speak of a body, I really mean a subset $B\subset \Bbb{R}^3$ (if you want to be technical then ok we can require this to be either Lebesgue measurable or Borel measurable). Next, we have a measure $m_B$ (defined on either the Lebesgue or Borel $\sigma$-algebra of $B$). Recall from above that measures should intuitively be thought of as "measurement devices telling how how big sets are"; also by indicating the subscript $B$, I'm trying to emphasize that this is very much dependent on the specific body we're interested in.
Almost every example you will encounter will be of this type: you start with some $\rho$ (or you have a word problem which tells you how to calculate $\rho$, eg you have a cone of total mass $M$, base radius $R$, height $H$, whose mass density increases linearly as you go radially outward) and from this you use the formula above to calculate integrals with respect to mass. The right-most integral is now a simple exercise in multivariable integration; you may have to parametrize the set $B$ appropriately, use symmetries carefully if applicable, use Fubini's theorem etc, or any other trick to evaluate the integral.
A measure space $(X,\mathcal{M},\mu)$. Here, $X$ is a set, $\mathcal{M}$ is a collection of "nice subsets" of $X$, and $\mu$ is what's called a measure (which is a function $\mathcal{M}\to [0,\infty]$). Very roughly, if $A\in\mathcal{M}$ (recall this means $A$ is a "nice" subset of $X$) then $\mu(A)$ is a number called the measure of $A$, and intuitively you should think of it as a "measurement device" telling you "how big" the set $A$ is.
A certain collection $\mathcal{L}^1(\mu)$ of "nice" functions $X\to\Bbb{R}$ (we can consider more general target spaces, but let's not bother now), called the Lebesgue-integrable functions on $X$, with respect to $\mu$. Most of the functions you encounter in an introductory physics class are going to be "nice enough" anyway, so I won't bother defining this space now.
A linear mapping $\mathcal{L}^1(\mu)\to\Bbb{R}$ which assigns to each "nice function" $f\in \mathcal{L}^1(\mu)$ a specific real number, denoted as $\int_Xf\, d\mu$ or $\int_Xf(x)\, d\mu_x$, called the Lebesgue integral of $f$ over $X$ with respect to the measure $\mu$ (again, I won't go into details of how this is carefully defined unless you really want me to). The $d$ here is just to make things look as classical as possible, don't read too much into this here.
Now given a body $B$, we can regard its moment of inertia $I=\int_Br^2\, dm$ using the Lebesgue integral as follows. When I speak of a body, I really mean a subset $B\subset \Bbb{R}^3$ (if you want to be technical then ok we can require this to be either Lebesgue measurable or Borel measurable). Next, we have a measure $m_B$ (defined on either the Lebesgue or Borel $\sigma$-algebra of $B$). Recall from above that measures should intuitively be thought of as "measurement devices telling how how big sets are"; also by indicating the subscript $B$, I'm trying to emphasize that this is very much dependent on the specific body we're interested in.
Almost every example you will encounter will be of this type: you start with some $\rho$ (or you have a word problem which tells you how to calculate $\rho$, eg you have a cone of total mass $M$, base radius $R$, height $H$, whose mass density increases linearly as you go radially outward) and from this you use the formula above to calculate integrals with respect to mass.
A measure space $(X,\mathcal{M},\mu)$. Here, $X$ is a set, $\mathcal{M}$ is a collection of "nice subsets" of $X$, and $\mu$ is what's called a measure (which is a function $\mathcal{M}\to [0,\infty]$). Very roughly, if $A\in\mathcal{M}$ (recall this means $A$ is a "nice" subset of $X$) then $\mu(A)$ is a number called the measure of $A$, and intuitively you should think of $\mu$ as a "measurement device" telling you "how big" the set $A$ is.
A certain collection $\mathcal{L}^1(\mu)$ of "nice" functions $X\to\Bbb{R}$ (we can consider more general target spaces, but let's not bother now), called the Lebesgue-integrable functions on $X$, with respect to $\mu$. Most of the functions you encounter in an introductory physics class are going to be "nice enough" anyway, so I won't bother defining this space now.
A linear mapping $\mathcal{L}^1(\mu)\to\Bbb{R}$ which assigns to each "nice function" $f\in \mathcal{L}^1(\mu)$ a specific real number, denoted as $\int_Xf\, d\mu$ or $\int_Xf(x)\, d\mu(x)$, called the Lebesgue integral of $f$ over $X$ with respect to the measure $\mu$ (again, I won't go into details of how this is carefully defined unless you really want me to). The $d$ here is just to make things look as classical as possible, don't read too much into this here.
Now given a body $B$, we can interpret its moment of inertia $I=\int_Br^2\, dm$ using the Lebesgue integral as follows. When I speak of a body, I really mean a subset $B\subset \Bbb{R}^3$ (if you want to be technical then ok we can require this to be either Lebesgue measurable or Borel measurable). Next, we have a measure $m_B$ (defined on either the Lebesgue or Borel $\sigma$-algebra of $B$). Recall from above that measures should intuitively be thought of as "measurement devices telling how how big sets are"; also by indicating the subscript $B$, I'm trying to emphasize that this is very much dependent on the specific body we're interested in.
Almost every example you will encounter will be of this type: you start with some $\rho$ (or you have a word problem which tells you how to calculate $\rho$, eg you have a cone of total mass $M$, base radius $R$, height $H$, whose mass density increases linearly as you go radially outward) and from this you use the formula above to calculate integrals with respect to mass. The right-most integral is now a simple exercise in multivariable integration; you may have to parametrize the set $B$ appropriately, use symmetries carefully if applicable, use Fubini's theorem etc, or any other trick to evaluate the integral.
Mathematical Setup
So far, you have learnt about Riemann integration: a function $f:[a,b]\to\Bbb{R}$ is Riemann-integrable if it is bounded and the supremum of lower Darboux sums equals the infimum of upper Darboux sums, in which case the integral is defined to be this common value, and is denoted as $\int_a^bf$. This definition generalizes almost verbatim to $n$-dimensions. Anyway, I want to highlight the structure of things: notice that you have three pieces of information:
- An interval $[a,b]$ (this is your starting setup).
- A collection, $\mathcal{R}_{[a,b]}$, of functions $[a,b]\to\Bbb{R}$, called the Riemann-integrable functions (these are the functions which are "nice enough" to be assigned an integral).
- A linear mapping $\mathcal{R}_{[a,b]}\to\Bbb{R}$ which assigns to each Riemann-integrable function $f$ its Riemann integral $\int_a^bf$ (which you can easily prove is a linear mapping).
I'm guessing you haven't taken a course in Lebesgue integration with repsect to arbitrary measures (because then this question becomes "almost obvious"). Let me try to outline the structure of this theory of integration, we again have three pieces of information
A measure space $(X,\mathcal{M},\mu)$. Here, $X$ is a set, $\mathcal{M}$ is a collection of "nice subsets" of $X$, and $\mu$ is what's called a measure (which is a function $\mathcal{M}\to [0,\infty]$). Very roughly, if $A\in\mathcal{M}$ (recall this means $A$ is a "nice" subset of $X$) then $\mu(A)$ is a number called the measure of $A$, and intuitively you should think of it as a "measurement device" telling you "how big" the set $A$ is.
A certain collection $\mathcal{L}^1(\mu)$ of "nice" functions $X\to\Bbb{R}$ (we can consider more general target spaces, but let's not bother now), called the Lebesgue-integrable functions on $X$, with respect to $\mu$. Most of the functions you encounter in an introductory physics class are going to be "nice enough" anyway, so I won't bother defining this space now.
A linear mapping $\mathcal{L}^1(\mu)\to\Bbb{R}$ which assigns to each "nice function" $f\in \mathcal{L}^1(\mu)$ a specific real number, denoted as $\int_Xf\, d\mu$ or $\int_Xf(x)\, d\mu_x$, called the Lebesgue integral of $f$ over $X$ with respect to the measure $\mu$ (again, I won't go into details of how this is carefully defined unless you really want me to). The $d$ here is just to make things look as classical as possible, don't read too much into this here.
As you can see, from a technical perspective, Lebesgue integration is much more involved, but structurally, it is essentially the same, there are three things at play: the general setup ($[a,b]$ vs $(X,\mathcal{M},\mu)$), the "nice collection of functions" ($\mathcal{R}_{[a,b]}$ vs $\mathcal{L}^1(\mu)$) and the integration mapping ($\int_a^b(\cdot)$ vs $\int_X(\cdot)\, d\mu$). The Lebesgue theory can be shown to be a generalization of the Riemann theory.
Using it in Physics
Now given a body $B$, we can regard its moment of inertia $I=\int_Br^2\, dm$ using the Lebesgue integral as follows. When I speak of a body, I really mean a subset $B\subset \Bbb{R}^3$ (if you want to be technical then ok we can require this to be either Lebesgue measurable or Borel measurable). Next, we have a measure $m_B$ (defined on either the Lebesgue or Borel $\sigma$-algebra of $B$). Recall from above that measures should intuitively be thought of as "measurement devices telling how how big sets are"; also by indicating the subscript $B$, I'm trying to emphasize that this is very much dependent on the specific body we're interested in.
For example, if $B$ is a solid cube of unit side-length, having uniform mass density of $\rho$, then by definition for any (Lebesgue/Borel measurable) subset $A\subset B$, we put $m_B(A):= \rho \cdot \text{volume}(A)$. So in this setting, the way $m_B$ measures size of a given set $A$ is relative to the density $\rho$ and the volume of the set $A$ in question. So, specializing some more, we have $m_B(B)=\rho \cdot \text{volume}(B)$; i.e the mass of the body in question is its density times its volume (of course, I defined things precisely to make this true).
Finally, we're considering the norm function $r:\Bbb{R}^3\to\Bbb{R}$, $r(p):= \lVert p\rVert:= \sqrt{p_1^2+p_2^2+p_3^2}$, i.e the Euclidean length of $p$. With this, the symbol $\int_Br^2\, dm_B$ makes perfect sense. We're integrating the function $r^2$ (which is a mapping $\Bbb{R}^3\to \Bbb{R}$, taking each $p\in\Bbb{R}^3$ to its squared length $r(p)^2 = p_1^2+p_2^2+p_3^2$) over the set $B$ with respect to the measure $m_B$. Thus, the symbol $\int_Br^2\, dm$ can be interpreted precisely using the concept of a Lebesgue integral.
A Few Remarks
Typically, one starts with a (Lebesgue/Borel measurable) set $B\subset \Bbb{R}^3$, and considers a function $\rho:\Bbb{R}^3\to[0,\infty)$ such that $\rho|_{B^c}=0$, i.e $\rho(x)=0$ for all $x\notin B$. This is what we refer to as the "density of $B$", so of course the interpretation is that for each $x\in\Bbb{R}^3$, $\rho(x)\geq 0$ is a number telling us the density of the body $B$ at the point $x$ (this is why if $x\notin B$, we require $\rho(x)=0$...because if there's is no body there, we should assign the density to be zero). From this density, we can construct a measure $m$ as follows: for any $A\subset \Bbb{R}^3$, we define $m(A):=\int_A \rho\, dV$ (where $dV$ stands for integration with respect to volume, or more precisely, the Lebesgue measure on $\Bbb{R}^3$).
Now, for any "nice" function $f:\Bbb{R}^3\to\Bbb{R}$, we can consider the Lebesgue integral $\int_{B}f\, dm$, and we can show that (if you carefully unwind the definition of Lebesgue integral, and use things like monotone convergence theorem) this equals $\int_{B}f\rho\, dV$, i.e \begin{align} \int_{B}f\,dm &= \int_{B}f\rho \, dV \equiv \int_{B}f(x)\rho(x)\, dV(x). \end{align} Physically, the way we interpret this equation is that integration of a function $f$ "with respect to mass" (which has a precise meaning as a Lebesgue integral) can be equivalently thought of as a "weighted integral" of $f$ times the density $\rho$ with respect to the usual volume integral. You may have encountered things like weighted integrals/ weighted averages in probability or some other context... well this is precisely that.
Almost every example you will encounter will be of this type: you start with some $\rho$ (or you have a word problem which tells you how to calculate $\rho$, eg you have a cone of total mass $M$, base radius $R$, height $H$, whose mass density increases linearly as you go radially outward) and from this you use the formula above to calculate integrals with respect to mass.
However, there are some examples where this is not the case. It is possible to have measures which cannot be "represented as a density". i.e there exist measures $m$ such that there is no $\rho:\Bbb{R}^3\to\Bbb{R}$ such that for all sets $A$, $m(A)=\int_A\rho\, dV$. The most famous of these examples is the dirac measure $\delta_0$, which assigns $\delta_0(A)=0$ if $0\notin A$ and $\delta_0(A)=1$ if $0\in A$. The mathematical way of saying this is that there are some measures on $\Bbb{R}^3$ which are not absolutely continuous with respect to the Lebesgue measure on $\Bbb{R}^3$, and hence have no Radon-Nikodym derivative with respect to Lebesgue measure.
Also, a question you may have thought of is "how to rigorously define the density of a body"; you may have seen something like $\rho = \frac{dm}{dV}$, and you may have wondered how is this defined precisely (this is obvious in some simple cases, but not always). Well, the answer again is related to the concept of Radon-Nikodym derivative.
Of course, this answer is definitely on the more mathematical side, but hopefully it gives you a few buzzwords to look up, and gives you something to look forward to in future more advanced analysis classes, where you can then relate the more abstract concepts in analysis down to something very physical.