In Royden's Real Analysis there's this generalization of Lebesgue's Dominated Convergence theorem (p.92):

Let $\{g_n\}$ be a sequence of integrable functions which converges a.e. to an integrable function g. Let $\{f_n\}$ be a sequence of measurable functions such that $|f_n| \leq g_n$ and $\{f_n\}$ converges to f a.e. If $$\int g = \lim \int g_n$$ then $$\int f = \lim \int f_n$$

Are there examples where the usual version, where there's just measurable g rather than the sequence $\{g_n\}$, isn't enough and it's necessary to use this generalization? I've never heard of any but it would be pretty interesting to find one.


Let's start out with a sequence of nonnegative integrable functions $g_n$ whose supremum is not integrable, but which converge a.e. to an integrable function $g$ and whose integrals converge to the integral of $g$. For example, on $[0,1]$ we might take $ g_n(x) = 1 + J_n(x)/x$ where $J_n$ is the indicator function of the interval $[1/(n+1), 1/n]$, and $g(x) = 1$.

Take any measurable $f_n \to f$ with $|f_n| \le g_n$ and converging a.e. to some measurable function $f$.

Note: I'm not claiming that it's "necessary" to use the generalization, just that it's not a completely obvious application of the standard LDCT.

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