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52
votes
9answers
6k views

Geometric proof of the Vandermonde determinant?

The Vandermonde matrix is the $n\times n$ matrix whose $(i,j)$-th component is $x_j^{i-1}$, where the $x_j$ are indeterminates. It is well known that the determinant of this matrix is $$\prod_{1\leq ...
18
votes
2answers
478 views

How many subsets of $[0,1)$ are there modulo null sets?

For subsets $A$ and $B$ of $[0,1)$, say $A\sim B$ iff $\lambda(A\Delta B)=0$ where $\lambda$ is Lebesgue measure. Question: How many equivalence classes of subsets of $[0,1)$ are there given AC? I ...
9
votes
1answer
730 views

Measure of a set of irrational numbers

Let $A$ be a set of all irrational numbers $\rho \in (0, 1)$ represented as a continued fraction $\rho=[a_{1}, a_{2},...,a_{n},...],$ such that $a_{n}\leq \text{const}\cdot n^{\epsilon}$ for some ...
4
votes
1answer
214 views

Uncountable atomless subalgebras of the Boolean algebra of all Jordan measurable sets in [0,1]

Definition: Suppose $\mathcal A$ is the Boolean algebra of all Jordan measurable sets in $I=[0,1]$ (i.e $\mathcal A=\{A\subseteq I: \mu(\partial(A))=0\}$, where $\mu$ is the Lebesgue measure and ...