let be a Lebesgue integrable function on [0,1], and let be a sequence of bounded measurable functions on [0,1] that converges uniformly to a function . Prove that
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help will be appreciated so much
i dont see why has to converge uniformly to . to me if , then (i am not sure how to show). and since there exists such that and since is integrable, so is .
by dominated convergence theorem, i think we get what we want.
can someone help me with this one?
Firstly, you will want to use the monotone convergence theorem. This gives us equality (your working with the Lebesgue integral and your lecturer is trying to show you how inadequate the Riemann integral is; the monotone convergence theorem doesn't work for Riemann, but it does for Lebesgue so they will tend to use it lots). You can use this here as you are in a compact interval and so a uniformly convergent sequence is non-decreasing.
So, can you prove that your sequence uniformly converges to if the sequence uniformly converges to ?
I'm now a tad confused. I was sure I found a result which said that any uniformly convergent sequence in a compact space is non-decreasing. However, I cannot seem to find this now and in retrospect it doesn't seem to make that much sense - something can converge uniformly from above...