= Solution
<Fubini's theorem> for the product of two <sigma-finite measure> spaces states that an integrable <measurable function> on the product has integrable sections <almost everywhere>, the section <integrals> define <Lebesgue integrable functions>, and both <iterated integrals> equal the product <integral>. The integrability hypothesis is <absolute integrability>. For nonnegative <measurable functions>, <Tonelli's theorem> gives equality of the possibly <infinite product> and <iterated integrals> without first assuming finiteness.
The displayed function is \b[not <Lebesgue integrable>]. Its value at the origin can be assigned arbitrarily, since a point is a <null set>. On $0<x<1$ and $0<y<x/2$,
$$
\left|\frac{x-y}{(x+y)^3}\right|
\ge\frac{x/2}{(3x/2)^3}=\frac4{27x^2}.
$$
Consequently <Tonelli's theorem> gives the divergent <lower bound>
$$
\int_0^1\int_0^1|f(x,y)|\,dy\,dx
\ge\frac2{27}\int_0^1\frac{dx}{x}=\infty.
$$
The signed <iterated integrals> illustrate why the hypothesis matters. For $x>0$, $f(x,y)=\partial_y[y/(x+y)^2]$, so <integration> first in $y$ and then in $x$ gives $1/2$. Integrating first in $x$ gives $-1/2$, using $f(x,y)=-\partial_x[x/(x+y)^2]$. Exceptional boundary sections have <measure> zero. Thus finite iterated values do not establish product integrability.
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