TITLE: A question about integral and equivalent measures
QUESTION [0 upvotes]: Let $\lambda$ be the Lebesgue measure and $\mu$ be a probability $\sigma$-finite measure on $[0, 1].$ Suppose $\lambda \ll \mu$ and $\mu \ll \lambda.$ What can we say about the  convergence of the integral $\int_{0}^{1}\log x\,d\mu$?

REPLY [0 votes]: I assume you want $\mu$ to be a probability measure on $[0,1]$ (all probability measures are $\sigma$-finite, because they are finite.)  In this case, the integral could converge or diverge.  If $\mu=\lambda$, then $\int_{0}^{1}\log x\,d\mu$ converges and equals $-1$.  On the other hand, if $\mu$ is given by the density function
$$\frac{d\mu}{d\lambda}=\frac{1}{x(1-\log x)^2}\qquad \text{on } [0,1],$$
then $\mu$ is a probability measure which is equivalent to $\lambda$, but
$\int_{0}^{1}\log x\,d\mu$ diverges to $-\infty$.
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