Let $E$ be a set of measure zero. I want to prove that the Cartesian product of $E$ and $R^n$ is also of measure zero.
I thought of maybe using intervals to cover the product, as I know $E$ is of measure zero, but I couldn't really work it out.
Prove that null set product $R^n$ is of measure zero
calculusmeasure-theorymultivariable-calculus
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Best Answer
I would use Tonellis theorem for non-negetive functions: $$\lambda^{m + n}(E \times \mathbb{R}^n) = \int_{E \times \mathbb{R}^n} d \lambda^{m+n}(x,y) = \int_{\mathbb{R}^n} \int_E d\lambda^m(x) d\lambda^n(y) = 0.$$