The Unapologetic Mathematician

Mathematics for the interested outsider

Positive and Negative Parts of Functions

Now that we have sums and products to work with, we find that the maximum of f and g — sometimes written f\cup g or [f\cup g](x)=\max(f(x),g(x)) — and their minimum — sometimes written f\cap g — are measurable. Indeed, we can write

\displaystyle\begin{aligned}f\cup g&=\frac{1}{2}\left(f+g+\lvert f-g\rvert\right)\\f\cap g&=\frac{1}{2}\left(f+g-\lvert f-g\rvert\right)\end{aligned}

and we know that absolute values of functions are measurable.

As special cases of this construction we define the “positive part” f^+ and “negative part” f^- of an extended real-valued function f as

\displaystyle\begin{aligned}f^+&=f\cup0\\f^-&=-(f\cap0)\end{aligned}

The positive part is obviously just what we get if we lop off any part of f that extends below 0. The negative part is a little more subtle. First we lop off everything above 0, but then we take the negative of this function. As a result, f^+ and f^- are both nonnegative functions. And if f is measurable, then so are f^+ and f^-. We can thus write any measurable function f as the difference of two nonnegative measurable functions

f=f^+-f^-

Conversely, any function with measurable positive and negative parts is itself measurable.

This is sort of like how we found that functions of bounded variation can be written as the difference between two strictly increasing functions. In fact, if we’re loose about what we mean by “function”, and “derivative”, we could even see this fact as a decomposition of the derivative of a function of bounded variation into its positive and negative parts.

It will thus be useful to restrict attention to nonnegative measurable functions instead of general measurable functions. Many statements can be more easily proven for nonnegative measurable functions, and the results will be preserved when we take the difference of two functions. Since we can write any measurable function as the difference between two nonnegative ones, this will suffice.

It will also be sometimes useful to realize that we may write the absolute value of a function as

\displaystyle\lvert f\rvert=f^++f^-

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May 7, 2010 - Posted by | Analysis, Measure Theory

9 Comments »

  1. [...] see this, first break up into its positive and negative parts and . If we can approximate any nonnegative measurable function by a pointwise-increasing sequence [...]

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  2. [...] First of all, from what we know about convergence in measure and algebraic and order properties of integrals of simple functions, we can see that if and are integrable functions and is a real number, then so are the absolute value , the scalar multiple , and the sum . As special cases, we can see that the positive and negative parts [...]

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  3. [...] an integrable function so that a.e., then is integrable. Indeed, we can break any function into positive and negative parts and , which themselves must satisfy a.e., and which are both nonnegative. So if we can establish [...]

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  4. [...] functions by the supremum above. General integrable functions overall are handled by using their positive and negative parts. Then you can prove the monotone convergence theorem, followed by Fatou’s lemma, and then the [...]

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  5. [...] general, we can break a function into its positive and negative parts and , and then [...]

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  6. [...] is, the upper variation of is the indefinite integral of the positive part of , while the lower variation of is the indefinite integral of the negative part of . And then we [...]

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  7. [...] last time. If is any measurable function, then its graph is measurable. Indeed, we can take the positive and negative parts and , which are both measurable. Thus all four sets , , , and are measurable. Choosing and we [...]

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  8. [...] in the sense that if either integral exists, then the other one does too, and their values are equal. As usual, it is sufficient to prove this for the case of for a measurable set . Linear combinations will extend it to simple functions, the monotone convergence theorem extends to non-negative measurable functions, and general functions can be decomposed into positive and negative parts. [...]

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  9. [...] going to need to assume that is nonnegative. We’d usually do this by breaking into its positive and negative parts, but it’s not so easy to get ahold of the positive and negative parts of in this case. [...]

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