# Tonelli's theorem using in mean residual life definition

If X is a nonnegative random variable representing the life of a component having distribution function F,the mean residual life is defined by

$$m(t) = E(X-t | X >t) = \frac{1}{\bar F(t)} \int_t^\infty (x-t) d\nu(x), t\geq 0$$ In papaer R. C. Gupta and D. M. Bradley (2003)" Representing the Mean Residual Life in Terms of the Failure Rate"mentioned that by writing $$x - t = \int_{t}^{x} du$$ and employing Tonelli's theorem yields the equivalent formula $$m(t) = \frac{1}{\bar F(t)}\int_t^\infty \int_t^x du d\nu(x) = \frac{1}{\bar F(t)}\int_t^\infty \int_u^\infty d\nu(x) du = \frac{1}{\bar F(t)}\int_t^\infty \bar{F}(u) du$$ How can we get this result by substituting the above integral and using Tonelli's theorem?

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$$\int_{-\infty}^\infty\int_{-\infty}^\infty 1_{[t,\infty]}(x)1_{[t,x]}(u)dud\nu(x)$$
To see that $\int_t^\infty\int_t^xdud\nu(x)=\int_t^\infty\int_t^xd\nu(x)du$, notice that the left integral is integrating over the region $x\geq u$ and $u\geq t$, in other words a triangle that starts at $t$, defined by $(u,x)$. The left hand side is integrating along vertical slices, while the right integral is going along horizontal ones
in $(u,x)$ coordinates, your triangle tip is at $(t,t)$, and when I say horizontal i mean keeping $x$ constant, and vice versa for vertical. –  Alex R. Oct 8 '12 at 5:59