So a well-known theorem from Lebesgue integration is the dominated convergence theorem. It talks about a sequence [itex]f_1,f_2,\ldots[/itex] of functions converging pointwise to a function [itex]f[/itex]. And if [itex]|f_n(x)| \leq g(x)[/itex] for an integrable function [itex]g[/itex], then we have [itex]\int f_n \to \int f[/itex].(adsbygoogle = window.adsbygoogle || []).push({});

But what if we have a given function [itex]f_\epsilon[/itex] which depends on some parameter [itex]\epsilon[/itex], which we are taking to zero? Suppose you've shown that [itex]f_\epsilon \to 0[/itex] pointwise as [itex]\epsilon \to 0[/itex]; however, you know [itex]f_\epsilon[/itex] is NOT bounded in [itex]\epsilon[/itex] - if you take [itex]\epsilon[/itex] large enough, you can make [itex]\sup_{x\in \mathbb R} f_\epsilon(x)[/itex] arbitrarily large. But, of course, we don't WANT to do that - we want to take [itex]\epsilon[/itex] small. And suppose you know that [itex]\epsilon < \epsilon_0[/itex] means [itex]\sup f_{\epsilon} < \sup f_{\epsilon_0}[/itex]. My question is this: Can we still apply the DCT to show that [itex]\lim_{\epsilon \to 0} \int_{\mathbb R} f_\epsilon = 0[/itex]? If so, why?

**Physics Forums - The Fusion of Science and Community**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Confusion regarding the dominated convergence theorem

Loading...

Similar Threads - Confusion regarding dominated | Date |
---|---|

B Confusion regarding area of this figure | Nov 5, 2016 |

Small confusion regarding logarithmic formula | Jan 29, 2015 |

Confusion regarding a proof for an infinite limit property. | Mar 13, 2013 |

Confusion regarding delta definition of limit | Mar 12, 2013 |

Confusion regarding differential forms and tangent space (Spivak,Calc. on Manifolds) | Nov 27, 2012 |

**Physics Forums - The Fusion of Science and Community**