Help understanding the definition of a specific measure.

Question

In an exercise, I am supposed to determine if a measure is finite or $\sigma$-finite, however, I do not understand the definition of said measure. The measure is defined as a "Borel measure on $[1,\infty)$ given by the density function, $1/x$, with respect to the 1-dimensional Lebesgue measure". Now, I know the definition of the Lebesgue measure, but what exactly is meant by a measure "given by the density function with respect to the Lebesgue measure"?

Answer 1

It means the measure $\mu$ is defined by $$ \mu(A) = \int_A\frac{d\lambda(x)}{x} $$ for any Borel $A\subseteq [1,\infty)$, with the integral being a Lebesgue integral.