I was looking at the proof of Wigner's Semicircle Law from this book..
Let $L_N$ be a random probability measure and $\sigma$ be a known probability measure supported in the interval $[-B,B]$. We wish to show for all $\epsilon>0$ and for any $f\in C_b(\mathbb{R})$, $$P(|\langle L_N,f\rangle -\langle \sigma,f\rangle|>\epsilon)\to 0$$ as $N\to \infty$. Suppose we have shown that $\lim\limits_{N\to\infty} P(\langle L_N,|x|^kI_{|x|>B}\rangle>\epsilon)=0$
Then the book says that it is enough to show the result for $f$ supported on $[-B,B]$ only.
I am having difficulties in realising why it is enough. Any help/hints!
Since $\Vert f \Vert < M$ for some $M>0$, we can find $k>0$ such that $B^k \geq M$, and $$f(x) \, I_{\vert x \vert > B} \leq M \leq B^k \leq \vert x \vert^k \, I_{\vert x \vert > B} \quad \mbox{ for any }x\in\mathbb R.$$ Thus, $$ P( \langle L_N , f \, I_{\vert x \vert > B} \rangle > \epsilon ) \leq P( \langle L_N , \vert x \vert^k \, I_{\vert x \vert > B} \rangle > \epsilon ),$$ and because $\langle \sigma, f \, I_{\vert x \vert > B} \rangle = 0$ we can write $$ P( \vert \langle L_n, f \rangle - \langle \sigma, f \rangle \vert > \epsilon ) \leq P( \vert \langle L_n, f \, I_{\vert x \vert \leq B} \rangle - \langle \sigma, f \, I_{\vert x \vert \leq B} \rangle \vert > \epsilon/2 ) + P( \vert \langle L_n, f \, I_{\vert x \vert > B} \rangle \vert > \epsilon/2 )$$ $$ \leq P( \vert \langle L_n, g \rangle - \langle \sigma, g \rangle \vert > \epsilon/2 ) + P( \langle L_N , \vert x \vert^k \, I_{\vert x \vert > B} \rangle > \epsilon/2 )$$ for $g$ supported in $[-B,B]$, and some $k>0$.