I am trying to understand the proof of existence of Haar Measure from Folland's A course in abstract Harmonic analysis.
Theorem : Every locally compact group $G$ possesses a left Haar Measure.
He used following proposition :
Let $\mu$ be a Radon Measure on the locally compact group $G$. Then $\mu$ is a left Haar measure if and only if $\int L_yf \,d\mu=\int f\,d\mu$ for every $f\in C_c^+(G)$ and for every $y\in G$.
So, it boils down to look for existence of radon measure with property defined above. We have Riesz representation theorem that gives a radon measure for a given positive linear functional on $C_c(G)$.
Riesz Representation theorem : Let $G$ be a locally compact hausdorff space and $\Lambda$ be a positive linear functional on $C_c(G)$. Then there exists a radon measure $\mu$ on $G$ such that $\Lambda(f)=\int f \,d\mu$ for every $f\in C_c(G)$.
So, it boils down to looking for existence of a suitable linear functional on $C_c(G)$.
How do I look for this linear functional. Proof in that books is confusing and unclear to me. Any suggestion on how to do this by approximation argument is most welcome.
Added : We have one one correspondence $$\{\text{Arbitrary functions }C_c^+(G)\rightarrow \mathbb{R}\}\rightarrow\prod_{f\in C_c^+(G)} \mathbb{R}.$$
Define the interval $X_f=[(f_0:f)^{-1},(f:f_0)]$ and we have seen that $I_{\varphi}(C_c^+(G))\subseteq X_f$. So, we have correspondence $$\{C_c^+(G)\xrightarrow{I_{\varphi}}\mathbb{R}\}\rightarrow \prod_{f\in C_c^+(G)} X_f\subseteq \prod_{f\in C_c^+(G)} \mathbb{R}.$$
We set $X=\prod_{f\in C_c^+(G)}X_f$. So, we identify maps $I_{\varphi}:C_c^+(G)\rightarrow \mathbb{R}$ by elements of $X$.
For a neighbourhood $V$ of $e$ we define $$K_V=\overline{\{I_{\varphi}\in X: \text{Supp}(\varphi)\subseteq V, \varphi\in C_c^+(G)\}}\subseteq X.$$ Here $I_{\varphi}$ has to be seen as an element of $X$ and as a linear functional by one one correspondence mentioned above.
As $X_f$ is compact for each $f$, $X$ is compact by Tychonoff's theorem. So, he some how proves that these collection $\{K_V\}$ has finite intersection property.
A space $X$ is compact iff any collection of closed subsets having finite intersection property have non empty intersection.
So, we see that $\bigcap K_V\neq \emptyset$. Let $I\in \bigcap K_V$. This is the linear functional that we are looking for.
As $I\in K_V=\overline{\{I_{\varphi}\in X: \text{Supp}(\varphi)\subseteq V, \varphi\in C_c^+(G)\}}$, given $\epsilon>0$ there is an element $I_{\varphi}$ with $\text{Supp}(\varphi)\subseteq V$ such that $|I-I_{\varphi}|<\epsilon$. It is not clear to me what $|I-I_{\varphi}|<\epsilon$ means when we see $I,I_{\varphi}$ as linear functionals.
He says $I\in K_V$ means that given any $\epsilon>0$ and any $f_1,\cdots,f_n\in C_c^+(G)$ there exists $\varphi\in C_c^+(G)$ such that $|I(f_j)-I_{\varphi}(f_j)|<\epsilon$ for all $1\leq j\leq n$.
Help me to understand why it is the case.