Show that a set is measurable w.r.t. product measure.;

by Math_QED   Last Updated November 10, 2018 14:20 PM

Let $f: (\Omega, \mathcal{F}) \to [0, \infty]$ be $\mathcal{F}$-measurable. Denote $\mathcal{R}$ for the Borel parts of the real numbers. Show that

$$A:= \{(t,\omega) \in \mathbb{R} \times \Omega\mid 0 \leq t < f(\omega)\} \in \mathcal{R} \otimes \mathcal{F}$$ (this is the product sigma algebra)

I defined $\phi: \mathbb{R} \times \Omega \to \mathbb{R}: (t, \omega) \to f(\omega)-t$.

It suffices to show that $\phi$ is measurable w.r.t. the product measure, since $A = \{\phi > 0\} \cap (\mathbb{R}^+ \times \Omega)$.

I have however trouble showing that $\phi$ is measurable.

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