Suppose that $f:\mathbb{C}\rightarrow \mathbb{C}$ is a non-constant entire function. By Liouville's theorem, we know that $f$ must take on arbitrarily large values. However Liouville doesn't say anything about what this large set must look like. In particular is it possible that the large values of $f$ are concentrated on a set of small measure?
More precisely, does there exist a non-constant entire function $f$ such that $\lambda(\{x: |f(x)|>1 \})<\infty$? Here $\lambda$ denotes the $2$-dimensional Lebesgue measure.
This set can have finite measure. See, for example, MR0537357 Golʹdberg, A. A. Sets on which the modulus of an entire function has a lower bound, Sibirsk. Mat. Zh. 20 (1979), no. 3, 512–518, 691. (There is an English translation: Siberian Math. Journal, 20 (1980) 360-364).