I'm writing notes for my upcoming class in Game Theory and I realized some time ago that I only need three properties of the Lebesgue measure $\lambda$ on $\mathbb{R^n}$.
- It is a non-negative measure on Borel subset of $\mathbb{R}^n$ that is strictly positive on open non-empty sets.
- Every proper hyperplane in $\mathbb{R}^n$ has measure zero.
- There is a polynomial function $P: (\mathbb{R}^n)^{n+1}\to \mathbb{R}$ such that: For all $(x_1,x_2,\ldots,x_{n+1})\in (\mathbb{R}^n)^{n+1}$ there is a one to one onto function $\ell: \{1,2,\ldots, n+1\}\to \{1,2,\ldots, n+1\}$ satisfying $$ \lambda(conv\{x_1,x_2,\ldots,x_{n+1}\}) =P(x_{\ell(1)},x_{\ell(2)},\ldots,x_{\ell(n+1)}) $$ where $conv$ stands for convex hull.
The Lebesgue measure $\lambda$ satisfies (3) by identifying the volume of a positively oriented simplex with determinant of vertices times a constant.
Q1. I am only interested however in the existence of a measure satisfying (1), (2), (3). Do I have to fully construct the cumbersome Lebesgue measure to do this?
Q2. Can I construct a measure satisfying (1) and (2), which measures bounded sets finitely, in a manner that is less cumbersome than the Lebesgue measure? (Thanks @5PM: Added this question in Edit)
Q3. Suppose instead of polynomial P I require that P be real analytic. Will a measure satisfying (1) (2) (3) always be Lebesgue absolutely continuous with an analytic density function (like the normal distribution on $\mathbb{R^n}$)? In other words, if I show the existence of a measure satisfying (1) (2) (3) have I simply developed the Lebesgue measure.