Event space (probability)

Question

My professor left us to solve this problem:

Let $\xi = \xi(\omega)$ and $\eta=\eta(\omega)$ be two random variables defined on the probability space $(\Omega, F, \mathbb{P})$.Show that $\{\omega: \xi(\omega) = \eta(\omega)\}\in F$.

Can somebody explain to me where I need to start? I understand that I need to show that these events belong to the event space but I don't have any idea how to do that.

Answer 1

Hint: consider the definition of random variable, then try to write the set $\{\omega: \xi(\omega) = \eta(\omega)\}$ in terms of sets you know are in $F$. The definition of sigma-algebra/sigma-field will be useful here.