Consider $\mathbb{R}^n$: let $B_n$ be the $n$-dimensional ball. There is a measure named $n$-dimensional volume, we can use it to calculate the $n$-volume of $B_n$. Denote this measure by $\mu_n$. Also there is a measure named surface area, we can use it to calculate the surface area of $(n-1)$-dimensional sphere $\mathbb{S}^{n-1}$ that is the boundary of $B_n$. Lets denote this measure by $\mu_{n-1}$.
In some sense we got 2 measures on $B_n$, one is $n$-dimensional $\mu_n(B_n) =\;$volume of $B_n$, and the other one is $(n-1)$-dimensional $\mu_{n-1}(B_n) =\;$surface area of $\;\partial B_n$.
My question is: are there any conventional $(\leqslant n-2)$-dimensional measures on $B_n$? For example is there something like a perimeter of a 3-ball (not volume or surface area)? Maybe all these measures just have to be trivial?
I guess we can always just intersect $B_n$ by $\mathbb{R}^{n-2}\subset\mathbb{R}^n$ through origin and calculate conventional $(n-2)$-volume of the section ball. But surface area $\mu_{n-1}$ is not like that: the surface area of a $(n-1)$-sphere is not equal to $(n-1)$-volume of $B_n$'s equator. I am unable to construct such new $(n-2)$-measure, but that doesn't mean no one can.
Also I thought about $\partial^2 = 0$: if $\mathbb{S}^{n-1} = \partial B_n$ has its measure then we can try to construct another one by taking another boundary, but since $\partial \mathbb{S}^{n-1} = \partial^2 B_n = 0$. And the only measure on the empty set is trivial one.