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Fixed typo in the Singer representation remark.
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Bokuan Li
2026-03-20 00:13:08 -04:00
parent 1d9f537b86
commit 780d5d362e
1 changed files with 2 additions and 2 deletions
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src/measure/radon/c0.tex
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@@ -176,9 +176,9 @@
\nu: \cb_X \to E^* \quad \dpn{y,\nu(A)}{E} = \int_{X \times B} \one_A(x) \cdot \dpn{y, \phi}{E} \mu(dx, d\phi) \nu: \cb_X \to E^* \quad \dpn{y,\nu(A)}{E} = \int_{X \times B} \one_A(x) \cdot \dpn{y, \phi}{E} \mu(dx, d\phi)
\] \]
It may be more tempting to use the strong formulation directly It may be tempting to use the strong formulation directly
\[ \[
\nu: \cb_X \to E^* \quad \dpn{y,\nu(A)}{E} = \int_{X \times B} \phi \cdot \one_A(x) \mu(dx, d\phi) \nu: \cb_X \to E^* \quad \nu(A) = \int_{X \times B} \phi \cdot \one_A(x) \mu(dx, d\phi)
\] \]
However, without additional assumptions on $E^*$, $\phi \cdot \one_A(x)$ may not be strongly measurable, which prevents this direct use of the Bochner integral. Thus the weak formulation is a necessary complication in the proof. However, without additional assumptions on $E^*$, $\phi \cdot \one_A(x)$ may not be strongly measurable, which prevents this direct use of the Bochner integral. Thus the weak formulation is a necessary complication in the proof.