What's the reason for the pre-image def. of Lebesgue measurable?

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What's the reason for the pre-image def. of Lebesgue measurable?

I.e. $f$ Lebesgue measurable if $\{ x : f(x) > c \in \mathbb{R}\}$ is measurable.

Like why is it significant that $f(x) > c$?