I have to compute the following integral:

$$ \lim_{k \rightarrow \infty} \int_{\mathbb{R}^n} \chi_{B_k} f \mathrm{d}\lambda_n(x) $$

$\chi_{B_k}(x) =\begin{cases} 1 &, \text{if } x \in B_k \\ 0 &, \text{otherwise} \end{cases}$, $B_k = \{ x \in \mathbb{R}^n \ | \ \| x \| \le k \}$ and $f \in \mathcal{L}^1 (\mathbb{R}^n)$.

$$f \in \mathcal{L}^1(\mathbb{R}^n) \Leftrightarrow \int_{\mathbb{R}^n} \max\{0,f\} \mathrm{d}\lambda(x) < \infty \text{ and } \int_{\mathbb{R}^n} \max\{0,-f\} \mathrm{d}\lambda(x) < \infty$$

My assumption is that $\lim \limits_{k \rightarrow \infty} \int_{\mathbb{R}^n} \chi_{B_k} f \mathrm{d}\lambda_n(x) = \int_{\mathbb{R}^n} f \mathrm{d}\lambda_n(x)$

But how to show? I tried to use the majorant criterion, but $|f| \in \mathcal{L}^1(\mathbb{R}^n)$ don't have to hold.

I appreciate any hint :) Thank you in advance.

  • $\begingroup$ But what are your assumptions on the function $f$? $\endgroup$ – Siminore Aug 15 '14 at 10:00
  • $\begingroup$ $f \in \mathcal{L}^1(\mathbb{R}^n) \Leftrightarrow \int_{\mathbb{R}^n} \max\{0,f\} \mathrm{d}\lambda(x) < \infty$ and $\int_{\mathbb{R}^n} \max\{0,-f\} \mathrm{d}\lambda(x) < \infty$ $\endgroup$ – DerJFK Aug 15 '14 at 10:04
  • $\begingroup$ @Siminore : above you see the definition. I also added it to the post. $\endgroup$ – DerJFK Aug 15 '14 at 10:24

When $f$ is non-negative, it is a consequence of the monotone convergence theorem.

For the general case, write $f=f^+-f^-$, where $f^+:=\max\{0,f\}$ and $f^-$ are integrable non-negative functions.

  • $\begingroup$ Is this right? $$ \lim_{k \rightarrow \infty} \int_{B_k} f_{\pm} \mathrm{d}\lambda(x) = \int_{\mathbb{R}^n} f_{\pm} \mathrm{d}\lambda(x) < \infty$$ $$ \lim_{k \rightarrow \infty} \int_{B_k} f \mathrm{d}\lambda(x) = \lim_{k \rightarrow \infty} \int_{B_k} f_+ \mathrm{d}\lambda(x) - \int_{B_k} f_- \mathrm{d}\lambda(x) = \lim_{k \rightarrow \infty} \int_{B_k} f_+ \mathrm{d}\lambda(x) - \lim_{k \rightarrow \infty} \int_{B_k} f- \mathrm{d}\lambda(x) = \int_{\mathbb{R}^n} f_+ \mathrm{d}\lambda(x) - \int_{\mathbb{R}^n} f_- \mathrm{d}\lambda(x) = ) = \int_{\mathbb{R}^n} f \mathrm{d}\lambda(x) $$ $\endgroup$ – DerJFK Aug 15 '14 at 10:58

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