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I have a question regarding the correct notation when integrating the Dirac distribution $\mu$. When treating it as a measure, I can write the Lebesgue inetgral $\int_{\mathbb{R}}\mu(dx)=1.$ What if I wish to treat $\mu$ as a distribution? Would I write $\int_{-\infty}^{\infty}d\mu(x)=1?$ If correct, what sort of integral would this be?


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Both are correct, however the more common notation is $\int_\mathcal{R} d\delta(x)=1.$ As a measure you consider its action against $C_c$ and so formally for all $f\in C_c(\mathcal{R})$, $\int_\mathcal{R} f(x)d\delta(x)=f(0)$. This is consistent with the definition of $\delta$ as a distribution; although it is not a function we may write $$\int_\mathcal{R} f(x)\delta(x)dx:=\langle \delta(x),f(x)\rangle=f(0)$$ and thus we can give a meaning to $\int_\mathcal{R} \delta(x)dx=1$.

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