what does $\frac{\text{d}x}{x}$ mean?

Question

I saw in a lecture recently the Gamma-function written like $$\Gamma (k) = \int_0^\infty e^{-x} x^k \frac{\text{d}x}{x}$$ and the professor said, that the integral was with respect to the measure $\frac{\text{d}x}{x}$. How is this meant? If I am given a set $A$ in $\mathbf{R}$, how could I for instance calculate $\frac{\text{d}x}{x}(A)$?

Thanks!

Answer 1

The measure $\frac{dx}{x}$ assigns the measure $\int_A \frac{1}{x} dx$ to a set $A$ (implicitly considered to be a subset of $[0,\infty)$ so that you guarantee positivity). Note that in this notation Lebesgue measure is sometimes denoted by just $dx$.