real analysis – How is it that bore sets of $mathbb{C}$ equal the boreal sets of $mathbb{R}^2$

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:

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