TITLE: Lebesgue Measure of $\mathbb{R} \times \{0\}$
QUESTION [0 upvotes]: I know this is probably obvious, and I know the answer is that it is (Lebesgue) measure zero, but I'm having a hard time wrapping my head around it. Looking for an intuitive explanation.
Question: What is the measure of: $\mathbb{R} \times \{0\}$, where $\times$ is the Cartesian product.

REPLY [0 votes]: Take the rectangles $[n,n+1]\times[-\varepsilon/2^{|n|},\varepsilon/2^{|n|}]$ for $n\in\mathbb Z$ and fixed $\varepsilon>0$. Then their area is $2\varepsilon/2^n$, and $\mathbb R\times\{0\}$ is contained in the union of all these rectangles. Summing up the areas gives $6\varepsilon$, so the (two)-dimensional Lebesgue measure of $\mathbb R\times\{0\}$ must be $\leq 6\varepsilon$. Since $\varepsilon$ was arbitrary, the result follows.
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