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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!!

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    To answer the first question: yes it's true, see the [fat Cantor set](http://en.wikipedia.org/wiki/Smith%E2%80%93Volterra%E2%80%93Cantor_set)2012-09-21

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