Average and aggregate values with a distribution function

Question

I'm reading an economics paper and I'm trying to understand if a statement made by the author is an assumption or the consequence of a previous definition.

The part I don't understand is the following:

At each period $t$ the state of the economy is described by the current distribution of wealth, represented by a distribution function $G_t(w)$ (which representes the fraction of the population with current wealth below $w$). Aggregate wealth, (which is also the average wealth) $W_t$ is given by

$W_t=\int wdG_t(w)$

I don't understand if in this case, with this distribution function, the aggregate wealth is always equal to the average wealth, or if it is an assumption. I also add some assumptions made before this statement, but I don't know if they're relevant.

After these first assumptions, I found this integral.

Closed economy with an infinite, discrete time horizion $t=0, 1, 2, \dots$ and a stationary population of infinitely-lived dynasties $I=[0,1]$. There are two goods [...].

Answer 1

Often economic models consider atomistic individuals with a mass of 1. This is without loss of generality in most cases. Aggregate wealth and average wealth are same in that case. Please check the model specification for the mass of individuals.

Answer 2

It's hard to believe that the aggregate wealth is equal numerically to the average wealth; this would in particular imply that there was only one person in the economy (certainly not a very interesting scenario!).

However, they are conceptually the same for a constant population: $n$ people sharing aggregate wealth W have average wealth $W/n$. Since they only differ by a constant, any reasonable operation you want to do to wealth will be unaffected whether you consider the average or the aggregate -- in particular, if the operation that you were using did care, then it would have to be specifically tailored to the units of value. But real systems are generally considered invariant to the units we use to measure them, so this would be a very bad thing.

My best guess is that when you read this author, you should think of these two quantities as representing the same concept, bearing in mind the constant population assumption; use whichever one makes more sense in the given context.