Existance of Haar Measure

Question

While showing the existence of Haar measure We consider all finite sequences of positive numbers $(c_i)_{i < n}$ and all finite sets $\{x_i\mid i < n\}⊂G$ Such that $f(x)\le \sum_i c_i g(x_i^{-1}x)$ for all $x∈G$ we define $$(f:g)=\inf\left\{\sum_i c_i \biggm|f(x)≤\sum_i c_i g(x_i^{-1} x) \right\} $$ Could you please elaborate on this definition any pictorial presentation of this definition, is there any analogue of this definition with definition of Lebesgue Measure

Answer 1

To get an intuitive feeling assume that $f$ is smooth and $\geq0$; furthermore assume that the "gauge function" $g:\ G\to{\mathbb R}\ $ is $\equiv1$ in a tiny neighborhood $U$ of $e\in G$, and is $\equiv0$ everywhere else. For given $x_k\in G$ the function $x\mapsto g(x_k^{-1} x)$ then is an analogous "rectangle function" in the tiny neighborhood $x_kU$ of the point $x_k$. A linear combination $$x\mapsto \sum_{k=1}^N c_k g(x_k^{-1} x)\tag{1}$$ with $c_k\geq0$ is then a "step function" on $G$ of total "$g$-volume" $\sum_{k=1}^N c_k$.

We now turn to $f$. It is pretty natural to define the "$g$-volume" $(f:g)$ of $f$ by the quoted expression: The function $f$ is approximated from above with step functions $(1)$, and one takes as $(f:g)$ the infimum of the so obtained "$g$-volumes".

It is, of course, a long story to grind out of this setup a quantity that is independent of any "gauge function" chosen a priori.