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From a pedagogical perspective, is there a way to explain to the Dominated Convergence Theorem using visual examples? This may sound tasteless for mathematicians but many times, engineering students do not have exposure to the ways of thinking that mathematicians are comfortable with.

I was hoping that with examples of plots of sequence of functions $f_n(x)$, there must be an easy way of explaining this beautiful result.

Your help and creative ways to explaining this theorem are appreciated.

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  • $\begingroup$ I just think of it as, as long as the area doesn't escape to infinite, then we can interchange limits and integrals. $\endgroup$ Jan 9, 2017 at 5:31
  • $\begingroup$ Not answering your question, but I usually get a lot of good ideas and intuitions by trying some examples and counter-examples such as here $\endgroup$
    – polfosol
    Jan 9, 2017 at 6:08

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I guess the best approach is to think of what can go wrong for non-dominated functions. A key counterexample to interchanging limits and integrals is if you can construct a sequence of functions that converges pointwise to 0 but all have constant area. So basically a sequence of functions that converges to an infinitely tall spike at zero with area 1. We can see immediately that a bounded sequence cannot have this problem.

However, nor can a dominated sequence. There can only be a finite amount of area under the dominating function, so the 'spike' cannot compensate to stay of constant area by getting taller as it thins out. Its area must eventually go to zero.

Addendum For visualization's sake, a good counterexample is $f_n(x) = nxe^{-\frac{1}{2}nx^2}$ for $x \ge 0.$

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Here's a simple example that you can easily visualize. $$ f_{n}(x) = \begin{cases} 1 & \text{if $x \in [n,n+1)$,} \\ 0 & \text{otherwise.} \end{cases} $$ The domain of $f_{n}$ is $[0,\infty)$.

There's no dominating function that is integrable, but the sequence converges to $0$ pointwise. As $n$ increases, the strictly positive part of $f_{n}$ escapes to $\infty$, and the dominating convergence theorem fails.

Now take any integrable function $g > 0$ on $[0,\infty)$. If we change the above example to $$ f_{n}(x) = \begin{cases} g(x) & \text{if $x \in [n,n+1)$,} \\ 0 & \text{otherwise,} \end{cases} $$ then the strictly positive part of $f_{n}$ asymptotically disappears (fast enough) before it escapes to $\infty$. So the dominating convergence theorem holds.

You can understand the role of a dominating function as a constraint that prevents the above escaping behavior.

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