Corollary 14.8.4`B9C312CB7069`

Corollary 14.8.4.B9C312CB7069 For two smooth functions $F,G$, the normalization constants satisfy

\[ -\sup _ X(G-F)\leq b(G)-b(F)\leq -\inf _ X(G-F). \]

In particular, $|b(G)-b(F)|\leq \| G-F\| _\infty $. If $G-F$ is nonconstant, both inequalities in the first display are strict.

Proof. Apply (14.8.2) along $F_ t=F+t(G-F)$ and integrate over $0\leq t\leq 1$. Each $\mu _{u_ t}$ is a probability measure with strictly positive smooth density, so the integral of a nonconstant continuous function against it lies strictly between the minimum and the maximum of that function. $\square$

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BibTeX citation

@misc{pome-tag-B9C312CB7069,
  author = {{Pome contributors}},
  title = {Corollary 14.8.4},
  howpublished = {Pome},
  url = {https://mingchenxia.github.io/Pome/tag/B9C312CB7069/}
}

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Corollary 14.8.4B9C312CB7069

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Corollary 14.8.4`B9C312CB7069`