Let $f(x):\mathbb{R}\to\mathbb{R}$ be a continuous strictly monotonic function. For example, $f(x)=\tanh(x)$.

Does the set $\{\mathbf{x} \in \mathbb{R}^n : \sum_i f(x_i) = 0 \}$ have Lebesgue measure zero?

If $f(x)$ is a linear function, then the set is a $n-1$ dimensional linear subspace of $\mathbb{R}^n$, for which the Lebesgue measure is zero. But what if $f(x)$ is nonlinear?

  • $\begingroup$ What methods do you know to show that a set has Lebesgue measure zero? Have you tried to find an answer for the case $n=1$ or $n=2$? $\endgroup$ – harfe May 25 at 19:35

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