Lebesgue Integration Proof

Asked May 14 '13 at 08:48

Active May 15 '13 at 06:12

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I have been trying to teach myself Lebesgue Integration and came across the following question in a textbook.

Let $g \subset [0,1]$ be bounded and measurable. Suppose that it satisifes the property $\int_{[0,1]} g \chi_{[0,c)}\, d\mu = 0$ for all $c \in [0,1]$. Show that $g=0$ a.e.

edited May 15 '13 at 06:12

asked May 14 '13 at 08:48

John

Why, does that answer not satisfactory? – Norbert May 15 '13 at 00:02
Note that $$\int_p^q gd\mu=\int_0^p gd\mu-\int_0^q gd\mu=\int_0^1 g\chi_{[0,p)}d\mu-\int_0^1 g\chi_{[0,q)}d\mu=0-0=0$$ – Norbert May 15 '13 at 00:08
So ask your question – Norbert May 15 '13 at 00:37
See my comment above where I have proved this – Norbert May 15 '13 at 00:49

Lebesgue Integration Proof

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