Fubinis Theorem

Fubinis Theorem states a possibility of changing order of integration in iterated integrals. The Theorem could not hold[^1] if the following conditions are not met

• The function $f(x,y)$ is continuous[^2]
• The function $f(x,y)$ is a bounded function[^2]

[^1]: Conversely, even if the given conditions are not satisfied Fubinis Theorem may still hold for a specific function. [^2]: Boundedness is not required; Discontinuities are allowed as long as $f$ offers absolute integrability over the region.

Riemann Version of Fubini’s Theorem

The Theorem states that if $f(x,y)$ is continuous on a closed, bounded rectangle $R\subset \mathbb{R}$

\iint\_R f(x,y),dA \= \int\_a^b \left( \int\_c^d f(x,y),dy \right) dx \= \int\_c^d \left( \int\_a^b f(x,y),dx \right) dy.