If $A$ is a set of positive measure say in $\mathbb{R}^2$ then $A$ does not necessarily have a rectangle of positive measure. This is true I suppose? Because we can apply iteration in Cantor fashion but the total measure can be less than zero by choosing the thrown away ratio small say $1/4$.
Similarly I have come across a question: Given three points forming a triangle in $\mathbb{R}^2$. Show that $A$ as above having positive measure in $\mathbb{R}^2$ has vertices similar to that triangle in $\mathbb{R}^2$? I am not saying that $A$ contains that triangle but it contains the vertices forming a triangle which is a similar triangle to the given any triangle? How to show this??
Thank you!!