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From: Jason Pawloski on 12 Oct 2006 01:20
Hello. This question is #38 in chapter 2 of Folland.

Suppose f_n -> f in measure and g_n -> g in measure. Show that f_n g_n
-> fg in measure if u(X) < infinity, but not necessarily if u(X) =
infinity.

I tried fooling around with |f_n - f| |g_n - g| >= |f_n| |g_n - g| +
|f| |g_n - g| = (|f_n| + |f|) |g_n - g|, so u({x : |f_n - f| |g_n -g|
>= epsilon}) <= u({x : (|f_n| + |f|)|g_n - g| >= epsilon}) and trying to get various other inequalities that I can "separate" to use the fact that f_n -> f and g_n->g in measure, but am getting no where. The "solutions" I found are worrisome since they do not use the fact that u(X) < infinity, which is especially troublesome since the next question asks to show an example where f_n g_n -/> fg where u(X) = infinity.

Can someone give me some insight on this problem? Thanks.

From: Jason Pawloski on 12 Oct 2006 01:39

Jason Pawloski wrote:
<snip>

Never mind, I got it.

Jason

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