I am reading the proof of Riesz Representation theorem(1.5.14) on Leon Simon's book:Geometric Measure Theory
And I got stuck at the following higlighted part. I completely understand the note he made befor the highlight part but I just cant reach his conclusion using that note. For your information: $$L(g)=\int_{X} g\cdot\nu d\mu$$ is proved .And $f \in C_c(X,[0,\infty))$, $X$ is locally compact and Hausdorff
Any help will be appreciated.
We want to find $g \in C_c(X, \mathbb{R}^n)$ with $|g| \leq f$ and $|L(g)|$ arbitrarily close to $\int_X f |v| \, d \mu$. Let us therefore look at $g_k = f\hat{g}_k$, since then we have $|g_k| \leq f$ from $|\hat{g}_k| \leq 1$. Next, by the triangle inequality we have
$\displaystyle \Bigg||L(g_k)| - \int_X f|v| \, d\mu \Bigg| \leq \Bigg|L(g_k) - \int_X f|v| \, d\mu \Bigg| \leq \int_X \big| g_k \cdot v - f|v| \big| \, d\mu$.
Now, using the trick that $|v| = \hat{v} \cdot v$ and the Cauchy-Schwarz inequality we get
$\displaystyle \int_X \big| g_k \cdot v - f|v| \big| \, d\mu = \int_X f|(\hat{g}_k - \hat{v}) \cdot v| \, d\mu \leq \int_X f|v||\hat{g}_k - \hat{v}| \, d\mu$.
Finally, it is mentioned earlier in your source that $v$ is a bounded function. The same is true of $f$ (being a continuous function of compact support). It therefore follows that $f|v|$ is bounded, and so by Hölder's inequality we get
$\displaystyle \int_X f|v||\hat{g}_k - \hat{v}| \, d\mu \leq \big|\big| f|v| \big|\big|_\infty \int_X |\hat{g}_k - \hat{v}| \, d\mu$
where the right hand side can be made arbitrarily small. We have thus shown what we wanted to show.