I`m unfamiliar with borel sets. I encountered a proposition and I want somebody to interpret and simplify it to a 101 probability theory sentence. the proposition is:
"The set of states of nature is taken to be $\mathbb{R}_+$. All random variables are measurable and real-valued functions defined on the measure space ($\mathbb{R}_+$,$\mathscr{B}$($\mathbb{R}_+$)) where $\mathscr{B}$($\mathbb{R}_+$) denotes the Borel sets of $\mathbb{R}_+$."
specially I wanna understand does this proposition impose some limits on pdf of related random variables? does any continuous/discrete random variable with continuous/discontinuous/differentiable/non-differentiable pdf or cdf satisfy this proposition?
thanks a lot!