# Can I always change the order of integration in an ordered multidimensional integral?

Imagine I have an integral of the following form:

$$I = \int_{-\infty}^{\infty} d\tau_1 \int_{\tau_1}^\infty d\tau_2 \int_{\tau_2}^\infty d\tau_3\ f(\tau_1,\tau_2,\tau_3) \tag{1}$$

Can I always commute the integrals, by changing the integration limits accordingly? And if not, when is it allowed/not allowed? For example:

$$I \overset{?}{=} \int_{-\infty}^\infty d\tau_2 \int_{-\infty}^{\tau_2} d\tau_1 \int_{\tau_2}^\infty d\tau_3\ f(\tau_1,\tau_2,\tau_3) \tag{2}$$

It seems to me that the region over which I integrate is the same, however I did run into discrepancies when numerically integrating $$(2)$$ vs. $$(1)$$ in some instances. I am not sure if they are artifacts from the numerical integration, hence the question.

• Thank you for your answer! Do you know if I can change the order of integration as above in the case where the integral is finite and $f(\tau_1,\tau_2,\tau_3) \geq 0$, i.e. $\left| f(\tau_1,\tau_2,\tau_3) \right| = f(\tau_1,\tau_2,\tau_3)$ (the integrand is real)? Is that a sufficient condition?
• As you can find here: en.wikipedia.org/wiki/…, if the integral is finite, and if $f$ is always non negative as in your assumptions, you can change the order. Commented May 25, 2020 at 11:03