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By Jensen's Inequality, we get | By Jensen's Inequality, we get | ||
| − | <math>\phi(\int_Afd\lambda \leq \int_A\phi(f)d\lambda</math> | + | <math>\phi(\int_Afd\lambda) \leq \int_A\phi(f)d\lambda = \frac{\int_A\phi(f)d\mu}{\mu(A)}</math> |
Revision as of 10:50, 22 July 2008
Define a function from the set of all measurable subset $ B $ of $ A $ as below
$ \lambda(B)=\frac{\mu(B)}{\mu(A)} $
This is clearly a measure on $ A $ with $ \lambda(A)=1 \frac{}{} $
Moreover, $ \int_{A}fd\mu = \mu(A)\int_A f d\lambda \frac{}{} $.
By Jensen's Inequality, we get
$ \phi(\int_Afd\lambda) \leq \int_A\phi(f)d\lambda = \frac{\int_A\phi(f)d\mu}{\mu(A)} $
