Question: Discuss whether or not the mapping (x mapsto|x|^{2}) (or its inverse image) is a homomorphism [ left(mathbb{R}^{d}, mathcal{B}left(mathbb{R}^{d}ight), Nleft(0, I_{d}ight)ight) xrightarrow{|x|^{2}}left(mathbb{R}, mathcal{B}(mathbb{R}), chi_{d}^{2}ight) ] of
Discuss whether or not the mapping \(x \mapsto\|x\|^{2}\) (or its inverse image) is a homomorphism
\[
\left(\mathbb{R}^{d}, \mathcal{B}\left(\mathbb{R}^{d}ight), N\left(0, I_{d}ight)ight) \xrightarrow{\|x\|^{2}}\left(\mathbb{R}, \mathcal{B}(\mathbb{R}), \chi_{d}^{2}ight)
\]
of probability spaces.
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