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I want to show that $\int \limits_{R^{n_1+n_2}}f = \int \limits_{R^{n_1}}\int \limits_{R^{n_2}} f(x, y) dy dx.$ In essence, I am trying to get the result in Tonelli's theorem. I know this is probably simple but I am stumped and can't seem to get started. Any tips?

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You didn't say what conditions apply to $f$.

$\mathbb{R}^n$ with the Lebesgue measure ($m_n$) is a complete measure space. If $f$ is integrable on $\mathbb{R}^{n_1+n_2}$, then Fubini says:

\begin{eqnarray} \int_{\mathbb{R}^{n_1} \times \mathbb{R}^{n_2}} f \, dm_{n_1+n_2} &=& \int_{\mathbb{R}^{n_1}} \left(\int_{ \mathbb{R}^{n_2}} f(x,y) dm_{n_2}(y) \right) dm_{n_1}(x) \\ &=& \int_{\mathbb{R}^{n_2}} \left(\int_{ \mathbb{R}^{n_1}} f(x,y) dm_{n_1}(x) \right) dm_{n_2}(y) \end{eqnarray}

$\mathbb{R}^n$ with the Lebesgue measure is a $\sigma$-finite measure space. If $f$ is measurable and non-negative, then Tonelli gives the same result.

(Implicit in the above is the fact that $m_{n_1+n_2} = m_{n_1} \times m_{n_2}$.)