Fatou's Lemma - Reverse Fatou Lemma

Reverse Fatou Lemma

Let f₁, f₂, . . . be a sequence of extended real-valued measurable functions defined on a measure space (S,Σ,μ). If there exists an integrable function g on S such that f_n ≤ g for all n, then

$\limsup_{n\to\infty}\int_S f_n\,d\mu\leq\int_S\limsup_{n\to\infty}f_n\,d\mu.$

Note: Here g integrable means that g is measurable and that .

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Famous quotes containing the word reverse:

“They shall beat their swords into plowshares, and their spears into pruninghooks: nation shall not lift up sword against nation, neither shall they learn war any more.”
—Bible: Hebrew Isaiah, 2:4.

The words reappear in Micah 4:3, and the reverse injunction is made in Joel 3:10 (”Beat your plowshares into swords ...”)