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I want to make more practice about how can find the integral with respect to counting measure and Dirac measure.

Any suggestion will be appreciated

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    Integral w.r.t. counting measure is summation. Maybe [this question](http://math.stackexchange.com/questions/764076/integration-with-respect-to-counting-measure) helps. As for Dirac measure, see [this question](http://math.stackexchange.com/questions/342803/integration-with-respect-to-dirac-measure) – 2017-02-07

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Let us start with the Dirac measure. Let $\delta_a(A)=0$ if $a\not\in A$ and $\delta_a(A)=1$ if $a\in A$. Then $$\int_X f\text{d}\delta_a=\int_{X\setminus\{a\}}f\text{d}\delta_a+\int_{\{a\}}f\text{d}\delta_a=0+\int_{\{a\}}f(a)\text{d}\delta_a=f(a)\int_{\{a\}}\text{d}\delta_a=f(a)$$

Now try to compute the integral wrt. the counting measure. Hint: $$\int_X f\text{d}(\mu+\nu)=\int_X f\text{d}\mu+\int_X f\text{d}\nu.$$

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    Thanks for that , but counting measure can't express it for two measures none of them counting measure. Is this right – 2017-02-07