Suppose $\hat{I}=[0,1]^\mathbb{N}$ is a Hilbert cube and $I=[0,1]$. Consider Lebesgue measures $m_1$ and $m_2$ on $\hat{I}$ and $I$ correspondingly. By Lebesgue measure on the Hilbert cube I mean the product of Lebesgue measures on the segments.

Let $f:\hat{I}\to I$ be a continuous surjection. Let $K$ be a Borel subset of $\hat{I}$ with $m_1(K) = 1$.

- Is it true that $m_2(f(K) ) = 1$ ?
- If the answer to the first question is no, what could be said if $f$ is a homeomorphism?
- If the answer to the first question is no, would it help if $f$ is Hölder continuous (with respect to metrics that are compatible with Borel structures)?

UPD: Question 2 should be read: "... if $f$ is an open map?"