For a test prep question:
Let $g(x)=\tanh (x)$, and let $\mu$ be the measure generated by $g$. Which subsets of the reals are $\mu$-measurable? Are polynomials integrable? And finally, is $|\sinh|$ integrable?
For a test prep question:
Let $g(x)=\tanh (x)$, and let $\mu$ be the measure generated by $g$. Which subsets of the reals are $\mu$-measurable? Are polynomials integrable? And finally, is $|\sinh|$ integrable?
Measurable sets are the same as the usual case because $\tanh$ is strictly increasing. Integration with function $f$ becomes $\int f(x)\tanh'(x) dx = \int \frac{f(x)}{\cosh^2(x)} dx$. Substituting a polynomial or $|\sinh|$ for $f$ will make the integral finite, so polynomials and $|\sinh|$ are integrable. (The domain is $(-\infty, \infty)$.)