Is every random variable bijective? Without getting too much into the measure theory behind this, my understanding is that a random variable maps from a sample space $\Omega$ to $\mathbb{R}$ (the random variable $X$ is defined as a function $X: \Omega \rightarrow \mathbb{R}$).
I have yet to come across a case in which each possible outcome in $\Omega$ does not have exactly one value in $\mathbb{R}$, but does this hold true in general? Does one need to delve deeper into the measure theory behind this to appropriately classify the bijectivity of random variables? Thank you!