= Solution
The <monotone convergence theorem> states that if <measurable functions> $0\leq f_n\uparrow f$ pointwise on a <measure space>, then $\int f_n\uparrow\int f$, allowing the limit to be infinite.
Extend each restriction to $I_n$ by zero on $I\setminus I_n$. The functions $f\mathbf1_{I_n}$ are nonnegative and increase pointwise to $f$ because the <intervals> are increasing and exhaust $I$. Hence
$$
\int_I f=\lim_{n\to\infty}\int_{I_n}f=\sup_n\int_{I_n}f.
$$
For nonnegative $f$, <integrability> means finiteness of this <integral>, so
$$
\boxed{f\in L^1(I)\iff\sup_n\int_{I_n}f<\infty.}
$$
No <boundedness> or <closedness> of the <intervals> is required.
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