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$\begingroup$

According to the book the following is true:

Let $f$ be a measurable function and suppose $|f| \leq h$ where $h$ is a real-valued function and $h$ is integrable. Then f is integrable.

My question is: suppose instead we have the inequality $ |f| \leq h$ but almost everywhere and of course $h$ assumed to be integrable. Is it still true that f is integrable?

Would this follow because the sets of measure zero "don't count" when integrating?

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    According to *what* book?2010-11-11
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    @Mariano: **The** book.2010-11-11
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    @Mariano: Fremlin's book.2010-11-12

1 Answers 1

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Yes on both counts.

Let $E$ be set of measure zero such that $|f|\leq h$ on $X-E$. Then let $g=f\chi_{X-E}$ (that is, redefine $f$ to be zero in $E$). Then $|g|\leq h$ everywhere, so $g$ is integrable. Then you have \begin{align*} \int_X g\,d\mu &= \int_{(X-E)\cup E}g\,d\mu = \int_(X-E)g\,d\mu + \int_E g\,d\mu\\ &= \int_{X-E}f\,d\mu + 0 = \int_{X-E}f\,d\mu = \int_{X-E}f\,d\mu + \int_E f\,d\mu\\ &= \int_{X}f\,d\,\mu, \end{align*} because $\int_E fd\,\mu$ exists and is equal to $0$, since $\mu(E)=0$ (so any sequence of simple functions that converges to $f$ will always have integral equal to $0$ over $E$).

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    Arturo, there is a "Misplaced \\".2010-11-11
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    @AD. Thanks; actually, it works when I'm in the preview, but complains as soon as I save the edits. Perhaps something to bring up in meta...2010-11-11
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    Ohh.. that is strange. Meta would be a good idea, and you could post the problematic Latex code there so they might fix it..??2010-11-11
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    @AD. Dang; it just happened again in a different post, this time with `\{` and `\\{`. I did write in meta: http://meta.math.stackexchange.com/questions/1115/different-behavior-in-parsing-latex-code-in-the-previewer-and-in-the-saved-post2010-11-11
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    I too have seen a problem with \{ they just disappeared! Was that the case for you?2010-11-11
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    Also see my answer in: http://math.stackexchange.com/questions/9833/if-i-t-1-exists-can-it-always-be-written-in-a-series-representation/9837#9837 Nothing worked!2010-11-11
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    @AD. See the question in meta; they changed the behavior of the renderer so that the escape characters are no longer working, but haven't gotten around to changing the behavior of the previewer. It's going to be annoying for a bit, I think.2010-11-11
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    @Arturo Magidin: I was told to use < instead of <. Soon we will be writing

    Here is a solution

    1+2<2+3

    ... :)
    2010-11-11
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    @AD.: Yes, `<` and `>` give problems because the renderer has to decide if they are meant as mathematical symbols or as HTML mark-up.2010-11-11
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    @Arturo: thank you very much.2010-11-11