Lebesgue Integral and Fubini Theorem

Lebesgue Integral:         $F(x)=\int_0^xf(y)dy$


Fubini Thorem:              $\int_a^b\int_c^df(x,y)dxdy=\int_c^d\int_a^b f(x,y)dydx$


If one day, for some reason, you need to calculate double integrals and the integral is a kind of Lebesgue Integral defined above. You simply want to change the order of integration. Then, you need to use Fubini theorem.  One simple example is given below:

$\int_0^1\int_0^xf(y)dydx=\int_0^1\int_y^1f(y)dxdy$

But, to be able to apply the Fubini theorem, you have to note that If the double integration is defined as $\int_a^b\int_c^df(x,y)dxdy=\infty$ then $\int_a^b\int_c^df(x,y)dxdy\ne \int_c^d\int_a^bf(x,y)dydx$. For more information and step by step example is given in this video.