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Wikipedia’s article on Dini’s Theorem states:

This is one of the few situations in mathematics where pointwise convergence implies uniform convergence; the key is the greater control implied by the monotonicity.

I’ve heard the word ‘control’ used like this a few times before, but I can’t get a good handle on what it means.

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up vote 4 down vote accepted

In mathematics, to control a mathematical object is to limit its behavior in an attempt to eliminate pathologies. Usually, what one does is to make a certain hypothesis on a mathematical object and then see what restrictions it imposes on the object.

In your case, the assumption of monotone convergence transforms pointwise convergence into uniform convergence. In general, the pointwise limit of a sequence of continuous $ \mathbb{R} $-valued functions on a topological space is a pretty wild animal. When the topological space is a metric space, we call such a function a ‘Baire-$ 1 $ function’. Baire-$ 1 $ functions can be hard to visualize. However, with the assumption of monotone convergence, the pointwise limit of a sequence of continuous $ \mathbb{R} $-valued functions on a compact space will always be a nice continuous $ \mathbb{R} $-valued function. One can thus say that the assumption of monotone convergence controls the behavior of pointwise limits.

Professor Terence Tao uses the word ‘control’ a lot because in almost all of his works, in order to prevent the branching complexity of his arguments from growing beyond his already-formidable powers of analysis, he usually has to make restricting hypotheses at certain stages.

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