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Let $f_n:\mathbb{R} \to \mathbb{R}, n \in \mathbb{N}_0$ be a sequence of nonnegative, Lebesgue measurable Functions with $\int f_n d\mu_L=1$. In Addition it holds $f_n \to f_0$ $\mu_L$-almost everywhere.

1.) Proof that $lim_{n \to \infty}\int |f_n-f_0|d\mu_L=0$.

2.) Conclude for the measures $\mu_n: \mathcal{L}(\mathbb{R})\to [0,\infty], \mu_n(B)=\int 1_{B_n}f_nd\mu_L$ $,n\in \mathbb{N}_0$ that for $n \to \infty$ $\sup_{B \in \mathcal{L}(\mathbb{R})}|\mu_n(B)-\mu_0(B)|\to0$ holds.

My Suggestions:

1.) $\lim_{n \to \infty}\int |f_n-f_0|d\mu_L$=$\lim_{n \to \infty}\int f_n-f_0+2(f_0-f_n)^+ d\mu_L$ $=\lim_{n \to \infty}\int f_n d\mu_L-\int f_0 d\mu_L+2\lim_{n \to \infty}\int0.5(|f_0-f_n|+(f_0-f_n))d\mu_L$ $=\lim_{n \to \infty}\int f_n d\mu_L-\int f_0 d\mu_L+\lim_{n \to \infty}\int|f_0-f_n|d\mu_L-\lim_{n \to \infty}\int f_n d\mu_L+\int f_0 d\mu_L$

What's the point from here now? Which theorem is needed? Is there a inequality for estimation?

2.) $sup_{B \in \mathcal{L}(\mathbb{R})}|\mu_n(B)-\mu_0(B)|=sup_{B \in \mathcal{L}(\mathbb{R})}|\int_{B}f_n d \mu_L -\int_{B}f_0 d\mu_L|=sup_{B \in \mathcal{L}(\mathbb{R})}|\int_{B}(f_n-f_0)d\mu_L|$ $\le sup_{B \in \mathcal{L}(\mathbb{R})}|\int_{B}|f_n-f_0|d\mu_L|$

Since these are trivial conversions, how does someone get the connection to 1.) in a right way? I don't know over what the Integral in 1.) is integrated. I would say now if there's a measure space $(X,\mathcal{A}(=\mathcal{L}(\mathbb{R})),\mu)$

$=sup_{B \in \mathcal{L}(\mathbb{R})}|\underbrace{\int_X |f_n-f_0|d\mu_L}_{=0}-\int_{X\setminus B} |f_n-f_0|d\mu_L|$.

I would say, that because of $B\subset X$ the 2nd integral is also $0$.

Help would be great

GE94
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  • There is a general theorem by Riesz that includes your problem as a particular case. The assumptions that $f_n,f_0\geq0$ simplify the problem greatly as the three line proof of Prof. Kavi Rama Murthy shows. The general case is interesting and a short proof is found here https://math.stackexchange.com/questions/51502/if-f-k-to-f-a-e-and-the-lp-norms-converge-then-f-k-to-f-in-lp – Mittens Jun 26 '20 at 20:01
  • I will have a look at it, the Theorem by Riesz hasn't been a topic yet, but soon. – GE94 Jun 27 '20 at 08:29
  • It is not a topic present, it appears in many books as a problem. – Mittens Jun 27 '20 at 11:00

1 Answers1

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$\int (f-f_n)^{+} \to 0$ by DC T with dominating function $f$. The first line in your attempt now shows that $\int |f-f_n| \to 0$.

$|\int_B f_n -\int_B f| \leq \int_B|f_n-f| \leq \int |f-f_n|$ for every $B$. Just take sup over $B$ and take limit.

Kavi Rama Murthy
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  • thanks for the answer. the notation $f_0$ was a little bit confusing and i forgot the $\lim$. – GE94 Jun 27 '20 at 08:28