I want to know if the following is true or not. If not please give me an answer what the left side is when computed the integral:
1 Let $\mu$ be the counting measure, then
$$\int 1_{\Omega}(x)\mathrm{d}\mu = \sum1_{\Omega}(x)$$
Edit: It is not an exercise. Just for understanding better.
Your notation does not make much sense to me. $\Omega$ is a set, $\xi$ a measure and thus $\xi(\Omega)\in\mathbb{R}$ and $\{\xi(\Omega)\}$ contains only one element/number. If $\lambda$ is the Lebenque-measure then $\lambda(\{\xi\Omega\})=0$. Also $\Omega=k$ does not make much sense if $\Omega$ is a set and $k\in\mathbb{R}$.
Edit: First one is correct, but only for the counting measure. More general case would be $$\int 1_{\Omega}(x)\mu(dx) = \int 1_{\Omega}d\mu = \mu(\Omega),$$ assuming that $\mu$ is a measure on the sigma-algebra $\mathcal{A}$ and $\Omega\in\mathcal{A}$.