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I'd like to show some properties of the Lebesgue integral.

I'd like to show that if $f$ is a simple function which is zero almost everywhere, then the Lebesgue integral $\int f(x) dx = 0$.

Similarly, I'd like to show this is also true for a measurable function $f$ which is zero almost everywhere.

I'm working through a real analysis textbook on my own, and I'm not quite sure what to do about this "almost everywhere." Do I have to consider separately a set of measure zero? Thank you as always.

Attempt for simple function:

If $f$ is a simple function that is zero almost everywhere, then $f = \sum_{i=1}^{n}a_{i}\chi_{E_i} = 0$.

Then, for each $i$, either $a_i = 0$ or else $m(E_i)= 0$.

By definition, $\int f(x) = \sum_{i=1}^{n}a_{1}m(E_i)$.

This summation is the sum of zeros. Thus, $\int f(x) = 0$ as desired.

Attempt for measurable function:

Assume $f$ is a non-negative measurable function that is zero almost everywhere.

By definition, $\int f(x) dx = \lim_{n \to \infty}\int f_n(x) dx$ where $\{f_n\}$ is a sequence of increasing, non-negative, simple functions that are all less than $f$.

Since $f$ is zero almost everywhere, then each non-negative, simple $f_n$ must also be zero almost everywhere.

Now, from above, we know that for each $n \in \mathbb{N}$, $\int f_n(x) dx = 0$.

Then, $\int f(x) dx = \lim_{n \to \infty} 0 = 0$.

Thus, $\int f(x) dx = 0$.

Now, for the general case...

We can write any measurable function $f$ in terms of its positive and negative parts. So, $f(x) = f^+(x) - f^-(x)$.

Both $f^+(x)$ and $f^-(x)$ are non-negative. Now if $f$ is zero almost everywhere, then both $f^+(x)$ and $f^-(x)$ are non-negative and zero almost everywhere.

Then, by above, $\int f^+(x) dx= \int f^-(x) dx= 0$

And by definition, $\int f(x) dx = \int f^+(x) dx - \int f^-(x) dx$.

So, $\int f(x) dx = 0 - 0 = 0$ as desired.

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    Welcome to Math.stackexchange! Here are some starters: A simple function has a range which is a finite set - they can be expressed in a 'simple' way. Can you tell us what that is? From there, you want to split it up into two parts - the non-zero part and the zero part. Then, tell us what your definition for the Lebesgue integral for a simple function is, and apply that definition.2012-05-13
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    Welcome to MSE! What book are using?2012-05-13
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    Some more hints: 1. Reduce to problem to characteristic functions. 2. What does it mean that $f$ is a characteristic funtion wich is $0$ almost everywhere? 3. How is defined the integral of a characteristic function?2012-05-13
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    I've added an attempt to the question. Any hints, advice, our direction is _much_ appreciated. Also, I'm using Krantz's Real Analysis and Foundations text. Thank you for your warm welcome. This looks to be a great community!2012-05-13
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    You have to assume that the simple function in the definition of the integral converge pointwise to $f$.2012-05-13

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