So I am kind of stuck here:
I need to show that $\delta_n \overset{w}{\rightarrow}\lambda$, where $\delta_n := \dfrac{1}{n} \sum_{i=1}^n \delta_{\frac{i}{n}}\left( (a,b) \right)$, where $\delta$ is the Dirac symbol and $\lambda$ the Lebesgue measure on $\left( \left[0,1\right], \vert \quad \vert\right)$.
I want to use Portmanteau to show that:
for $(a,b) \subset \left[ 0,1 \right]$, it holds that $\liminf_{n\rightarrow \infty}\dfrac{1}{n} \sum_{i=1}^n \delta_{\frac{i}{n}}\left( (a,b) \right) \geq \lambda\left( (a,b) \right)$.
Intuitively it should work, as it seems to approximate the Riemann Sum of a function $f\in \mathcal{C}_b\left( [0,1] \right)$, but I have a hard time formalizing the proof.
Any hint/help appreciated, thank you.