This is one of those things that I'm pretty sure is true (and vaguely recall the statement from a class or textbook) but can't seem to find quick verification on the internet.
Say we have a measure space $(\mu, \mathbb{B},X)$. Let $f\geq 0$ be a measurable function. If we define $\lambda(E)=\int_E f\,d\mu$ for $E\in\mathbb{B}$, then is $\lambda$ another measure on $(\mathbb{B},X)$?
I guess above, I should say, if $\mu$ is a positive measure, then is $\lambda$ also a positive measure the way it's defined?
What if $f$ isn't necessarily nonnegative? We clearly won't get a positive measure $\lambda$, but is $\lambda$ still a signed measure?