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I know there are many proofs for this but I am having trouble proving this fact using my book's definition.

My book defines first a non negative measurable function f as a function that can be written as the limit of a non decreasing sequence of non-negative simple functions.

Then my book defines that a function (taking on both positive and negative values) is measurable if both its positive part and negative part are measurable.

Let f and g be 2 measurable functions. Show that f + g is meaasurable as well.

If f and g are both non-negative, then it is clear that if $s_n$ is a nondecreasing sequence converging to f and $t_n$ is a nondecreasing sequence converging to g, then $s_n + t_n$ is a nondecreasing sequence converging to f + g.

However, I am having trouble with the more general case where f and g can take both positive and negative values. I am trying to show that the positive part of (f + g) is measurable, which means there exists a nondecreasing sequence of non-negative simple functions converging to $(f + g)^+$.

If I choose an x where f(x) and g(x) are non-negative, then it is clear how to construct such a sequence.

However, if I choose an x where f(x) > 0 and g(x) < 0 and f(x) + g(x) > 0, then I can represent this as $f^+ - g^-$. but i can't seem to come up with a sequence of functions converging to $f^+ - g^-$ AND is non decreasing. If $s_n$ is a nondecreasing sequence converging to $f^+$ and $t_n$ is a nondecreasing sequence converging to $g^-$, $s_n - t_n$ converges to $f^+ - g^-$ but it may not be a non decreasing sequence.

Can someone help me with this problem?

My book defines first a non negative measurable function f as a function that can be written as the limit of a non decreasing sequence of non-negative simple functions.

Then my book defines that a function (taking on both positive and negative values) is measurable if both its positive part and negative part are measurable.

Let f and g be 2 measurable functions. Show that f + g is meaasurable as well.

If f and g are both non-negative, then it is clear that if $s_n$ is a nondecreasing sequence converging to f and $t_n$ is a nondecreasing sequence converging to g, then $s_n + t_n$ is a nondecreasing sequence converging to f + g.

However, I am having trouble with the more general case where f and g can take both positive and negative values. I am trying to show that the positive part of (f + g) is measurable, which means there exists a nondecreasing sequence of non-negative simple functions converging to $(f + g)^+$.

If I choose an x where f(x) and g(x) are non-negative, then it is clear how to construct such a sequence.

However, if I choose an x where f(x) > 0 and g(x) < 0 and f(x) + g(x) > 0, then I can represent this as $f^+ - g^-$. but i can't seem to come up with a sequence of functions converging to $f^+ - g^-$ AND is non decreasing. If $s_n$ is a nondecreasing sequence converging to $f^+$ and $t_n$ is a nondecreasing sequence converging to $g^-$, $s_n - t_n$ converges to $f^+ - g^-$ but it may not be a non decreasing sequence.

Can someone help me with this problem?

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