I have a very basic question about measure theory (I have just started learning it). Consider the unit interval, with the usual Borel sigma algebra $\mathcal{B}.$ In other words, $\mathcal{B}$ is the sigma algebra generated by $\tau,$ where $\tau$ is the subspace topology on $[0,1],$ which is induced by the Euclidean topology on $\mathbb{R}.$
I am hoping that somebody can help me characterize the measurable functions from some measurable space $(\Omega, \Sigma _\Omega)$ to $[0,1]$ (with the aforementioned Borel sigma algebra $\mathcal{B}$). I saw a somewhat similar problem posed in a previous question, but in that case they were interested in measurable functions to the reals. Anyway, that answer caused me to conjecture as follows:
A function $f$ from $\Omega$ to $[0,1]$ is a measurable function from $(\Omega, \Sigma _\Omega)$ to $([0,1], \mathcal{B})$ if and only if for each $x \in [0,1]$ we have that $f^{-1} \left( [0,x] \right) \in \Sigma _\Omega .$
I was hoping somebody could tell me if this conjecture is correct, or else how to characterize when a function from $\Omega$ to $[0,1]$ is measurable.
