If $\varepsilon_a(dx)$ is the point mass at point $a\in\mathbb{R}$, I want to calculate $E_a[X]$ where $X$ is a real valued random variable and $E_a$ is expectation with respect to $\varepsilon_a(dx)$.
I'm not sure if my understanding is correct:
$$E_a[X]=\int_{\Omega} X\cdot\varepsilon_a(dx)=\int X1_{\{a\}}(x)dx=a$$
i.e. only when X takes the value of $a$ does the integrand be anything other than 0. Since $a$ is a point on real line, there is only one such point?
Let $\Omega=\mathbb R$ (otherwise the question makes no sense). For every $a$ in $\Omega$ and every random variable $X:\Omega\to\mathbb R^n$, by definition of the Dirac measure $\delta_a$, $$ \int_\Omega X\,\mathrm d\delta_a=\int_\Omega X(\omega)\,\mathrm d\delta_a(\omega)=X(a). $$