Suppose a country where the distribution of income $Y$ is given by a Probability Density Function (PDF) that we call $f_Y(y)$, obviously:
$$\int_0^\infty f_Y(y)\ \text{d}y = 1$$
The expected proportion of accumulated income $p_{acc}(y)$ by people with income below $y$ is given by:
$$0 \le p_{acc}(y) = \frac{\int_0^y y' f_Y(y')\ \text{d}y'}{\int_0^{\infty} y'f_Y(y')\ \text{d}y'} \le 1$$
I am almost sure that in the general case, outside of income distribution problems, this type function $p_{acc}(y)$ has a generic name, but I can't figure out which one. That is, given any PDF $f_Y$, I don't know if anyone knows what this computable magnitude is called for any type of PDF.
This is the Lorenz curve, whose integral is the Gini coefficient.