2.5 Corollary. A function $f: X rightarrow mathbb{C}$ is $mathcal{M}$ -measurable iff $operatorname{Re} f$ and

$operatorname{Im} f$ are $mathcal{M}$-measurable.Proof. This follows since

$mathcal{B}_{mathbb{C}}=mathcal{B}_{mathbb{R}^{2}}=mathcal{B}_{mathbb{R}}

otimes mathcal{B}_{mathbb{R}}$ by Proposition $1.5 .$

In Corollary 2.5, why are we allowed to say that $mathcal{B}_{mathbb{C}}=mathcal{B}_{mathbb{R}^{2}}$? How could a set of complex numbers be in $mathcal{B}_{mathbb{R}}$? Is this a different notion of equality used here?

The following are referenced propositions: