In general, if and are topological spaces is continuous iff for all . But ( -algebra generated by ) so, every continuous function is Borel measurable.
I have a question regarding the definition of Borel Measurable Functions.
Borel Measurable Functions is defined as the smallest collection of real-valued functions on R that contains the collection of continuous functions and is closed under pointwise limits.
But we know that the collection of continuous functions is NOT closed under pointwise limits. So when defining a collection of functions that IS closed under pointwise limits, should we exclude the collection of continuous functions? Does this definition seem contradictory? or am I missing something? (I know it's probably the latter.)
Any input is appreciated, thanks in advance.