Conditions for Fubini's theorem in simple terms

Question

Could someone explain the conditions required to satisfy Fubini's theorem in layman's terms, without going into measure theory? I'm a high school student trying to gain a simple understanding of when Fubini's theorem doesn't apply, but everything I've found online is in terms of measure spaces. Would it be possible to explain the conditions in terms of continuity and boundedness over the region of integration?

Answer 1

Another version of Fubini's theorem states that if $A$ and $B$ are $\sigma$-finite measure spaces, not necessarily complete, and if either $$ \int_{A}\left(\int_{B}|f(x, y)| \mathrm{d} y\right) \mathrm{d} x<\infty \quad \text { or } \quad \int_{B}\left(\int_{A}|f(x, y)| \mathrm{d} x\right) \mathrm{d} y<\infty $$ then $$ \int_{A \times B}|f(x, y)| \mathrm{d}(x, y)<\infty $$ and $$ \int_{A}\left(\int_{B} f(x, y) \mathrm{d} y\right) \mathrm{d} x=\int_{B}\left(\int_{A} f(x, y) \mathrm{d} x\right) \mathrm{d} y=\int_{A \times B} f(x, y) \mathrm{d}(x, y) $$ In this version the condition that the measures are $\sigma$-finite is necessary.