The Unapologetic Mathematician

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

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

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

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