Corollary

Tag 08OJ, permanent. Cite it as [Bourbaki, Tag 08OJ].

— Let $\lambda \in \mathscr{M}(X; \mathbf{R}), \ \mu \in \mathscr{M}(Y; \mathbf{R})$ . Then

\begin{cases} (\lambda \otimes \mu)^+ = \lambda^+ \otimes \mu^+ + \lambda^- \otimes \mu^- , \\ (\lambda \otimes \mu)^- = \lambda^+ \otimes \mu^- + \lambda^- \otimes \mu^+ . \end{cases}

For, by virtue of Prop. 3,

\begin{align*} (\lambda \otimes \mu)^+ &= \frac{1}{2} (\lambda \otimes \mu + |\lambda| \otimes |\mu|) \\ &= \frac{1}{2} ((\lambda^+ - \lambda^-) \otimes (\mu^+ - \mu^-) + (\lambda^+ + \lambda^-) \otimes (\mu^+ + \mu^-)) \\ &= \lambda^+ \otimes \mu^+ + \lambda^- \otimes \mu^- . \end{align*}

The argument for $(\lambda \otimes \mu)^-$ is similar.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

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