Hey guys I have a question on the Borel measurability of this set $\{{(x,y):x∈E,0<y<f(x)}\}$ when $f$ is a continuous function defined in an open set. Can anyone help me out? I think the set is open, but I'm a bit confused with the proof. Any help is appreciated thanks.
2026-03-25 15:40:12.1774453212
Borel measurability of a set
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Your instinct that the hypograph is open is correct. To see this, define $F: E \times \mathbb{R} \to \mathbb{R}$ by $F(x,y) = f(x) - y$. Since $f$ is continuous, so is $F$. Your set is the pre-image of $(0, \infty)$ and is therefore open.