I've been given a combined assignment, in which it's desired that I find the set $\left \{ u \geq a \right \}: \forall a\in \mathbb{R}$ with the function $u(t)=\left\{\begin{matrix} 3, t<0,\\ t, t\geq 0. \end{matrix}\right.$
My solution being. For $u:\mathbb{R} \rightarrow \mathbb{R}$. $\left\{ (u,t) \in \mathbb{R}, a \in \mathbb{R}: u\geq a..iff.. \forall t: u\geq 0 \wedge t\in(0,3]\cup(-\infty,0):3\geq u\geq0 \wedge \forall t>3:u>3 \right \}$ I know this is a very messy description. but I've found it might just describe it sufficiently. Of course if anyone has recommendations, please share.
My actual question goes to the next part of this assignment, where I'm asked to show that $u$ is $\mathcal{B}(\mathbb{R})/\mathcal{B}(\mathbb{R})$-measurable. I've come to understand that to show this, I'd have to show that $u^{-1}(B) \subset \mathcal{B}(\mathbb{R}), \forall B \subset \mathcal{B}(\mathbb{R})$. So I'd have to find $u^{-1}$ and than show that for a subset of $\mathcal{B}(\mathbb{R})$ the above is true.
I guess my question for you guys than is, 1. is my definition of $\left \{ u \geq a \right \}: \forall a\in \mathbb{R}$ correct in your eyes and 2. how do I show $u^{-1}(B) \subset \mathcal{B}(\mathbb{R})$
Best regards and thanks in advance