Show that a function over a mesurable set is the center of mass function

Question

First of all excuse me because my english is not good enough. I need some help with this excercise, I tried to solve it for 1 hour but nothing ocurred. I know you are very altruist. Thank you.

Answer 1

the relation for two disjoint measurable sets is easily extended to any finite number of such sets. If the sets are $A_j$, with union $A$ then: $$ \mu(A) r(A) = \sum_{j=1}^n \mu(A_j) r(A_j) $$ with $$ \mu(A) = \sum_{j=1}^n \mu(A_j) $$

now apply this to a unit-sided $p$-dimensional hypercube, which can be regularly subdvided as finely as desired.

this establishes the result for finite-dimensional hypercubes. perhaps you can take the argument a little further.