Consider the function $u:\mathbb{R}\to [0,\infty]$ given by $$ u(x)=\sum_{n=1}^{\infty}\frac{1}{n^{2}}1_{[n,n+1]}(x) $$ I have determined that $\int_{\mathbb{R}}u\, \mathbb{d}\lambda=\pi^{2}/6$ where $\lambda$ is a Lebesgue measure. I want to determine the integrals $\int_{\mathbb{R}}u\, \mathrm{d}\delta_{3}$ and $\int_{\mathbb{R}}u\, \mathrm{d}\delta_{\pi}$, where $\delta_{y}$ is a Dirac measure in point $y\in \mathbb{R}$.
What I did: first, we have $$ \int_{\mathbb{R}}u\, \mathrm{d}\delta_{3}=u(3)=\sum_{n=1}^{\infty}\frac{1}{n^{2}}1_{[n,n+1]}(3). $$ Since $1_{[n,n+1]}(3)=1$ where $3\in[n,n+1]$ is satisfied if $n=\lbrace 2,3\rbrace \subset \mathbb{N}$. That is, $1_{[2,3]}(3)=1_{[3,4]}(3)=1$. Therefore $$ \sum_{n=1}^{\infty}\frac{1}{n^{2}}1_{[n,n+1]}(3)=\frac{1}{2^{2}}+\frac{1}{3^{2}}. $$ Similarly $$ \int_{\mathbb{R}}u\, \mathrm{d}\delta_{\pi}=u(\pi)=\frac{1}{3^{2}}. $$ Are they correct?