= Solution
The indicated <measurable function> is nonnegative. <Tonelli's theorem> applies without first assuming <integrability> and gives
$$
\int_{\mathbb R^2}h=\int_a^b\left(\int_1^\infty x^{-y}\,dx\right)dy
=\int_a^b\frac{dy}{y-1}=\log\frac{b-1}{a-1}<\infty.
$$
Thus $h$ is integrable. <Fubini's theorem> now allows the reverse order. For $x>1$,
$$
\int_a^bx^{-y}\,dy=\frac{x^{-a}-x^{-b}}{\log x}.
$$
Consequently
$$
\boxed{\int_1^\infty\frac{x^{-a}-x^{-b}}{\log x}\,dx=\log\frac{b-1}{a-1}.}
$$
At $x=1$ the quotient has the removable limit $b-a$; that endpoint does not create an additional divergence.
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