measure theory - Dimension of the measurable space $mathbb{R}^n$

Consider $mathbb{R}^n$ as measurable space with the Borel algebra. If $mathbb{R}^n$ and $mathbb{R}^m$ are isomorphic (in the category of measurable spaces, i.e. there are measurable maps in both directions, which are inverse to each other), can we conclude $n=m$? Note that this statement is stronger than the invariance of dimension in topology and I doubt that it is true. Can you give a counterexample?

• Next at.algebraic topology - Ring of closed manifolds modulo fiber bundles
• Previous dg.differential geometry - Cobordisms of bundles?