Is there a name for this class of k-homogeneous measures?

Question

For a subset $S\subseteq\mathbb{R}^2$ write $$ cS = \{ cs\; |\; s\in S \}, $$ and define $F_c$ for a measure $\mu$ as $$ F_c\mu(\Omega) = \mu(c\Omega) $$ for all $\Omega\in\sigma(\mathbb{R}^2)$, where $\sigma(\mathbb{R}^2)$ are the Borel sets of $\mathbb{R}^2$. I am looking for information about the following set of measures $$ \mathcal{M}(k) = \{ \mu\in\mathcal{M} \; | \; \forall c\in\mathbb{R},\Omega\in\sigma(\mathbb{R}^2):\, F_c\mu(\Omega)=c^k\mu(\Omega) \}. $$ Usually a function that satisfies $$ f(cx) = c^kf(x) $$ is called $k$-homogenous. So I am tempted to call $\mathcal{M}(k)$ the set of $k$-homogenous measures. However, I have not been able to find information about this set (I keep ending up at homogenous metric spaces). Is there a name for this set so that I can look find more information about them? Or is there a book about this kind of measures?

Answer 1

I am not sure if there exists literature on this definition, yet I think the family of these kinds of measures should be quite restrictive. Let us denote $A=\mu(B(0,1))$, where $B(0,1)$ is the ball centered at $0$ with radius $1$ .For natural numbers $n$ $D(n):=B(0,\frac{1}{2^n})\setminus B(0,\frac{1}{2^{n-1}})$ and denote $B=\mu(D(0))$, furthermore note that $D(n)=2D(n+1)$. Then we have, \begin{align*} A=\mu(B(0,1))&=\sum_{n}\mu\left(B(0,\frac{1}{2^n})\setminus B(0,\frac{1}{2^{n-1}})\right)\\ &=\sum_{n}\mu\left(D(n)\right)\\ &=\sum_{n}\mu\left(2^{-n}D(0)\right)\\ &=B\sum_{n}(2^{-n})^k\\ &=(1-\frac{1}{2^k})A\sum_{n}(2^{-n})^k \end{align*} Showing that $1=2^k\left(\sum_{n}(2^{-n})^k\right)$; does this equation in $k$ have many solution?