How many Borel-measurable functions from $\mathbb{R}$ to $\mathbb{R}$ are there?

Question

How many Borel-measurable functions from $\mathbb{R}$ to $\mathbb{R}$ are there? The motivation is from this answer of mine on MathOverflow.

Answer 1

Sune, there are as many Borel-measurable functions as there are reals.

It is easy to see that a function is Borel-measurable iff the preimage of each open interval with rational end-points is a Borel set. Moreover, the function is completely determined by the sequence of pairs $(I,B)$ where $I$ varies over the intervals and $B$ is the preimage of $I$.

There are countably many possible $I$ and to each corresponds one of ${\mathfrak c}=|{\mathbb R}|$ many possible Borel sets. The total number of Borel functions is then bounded above by $|{\mathbb R}^{\mathbb N}|={\mathfrak c}$. Since each constant function is Borel-measurable, ${\mathfrak c}$ is also a lower bound.