Definition talk:Negative Part

Is this defined correctly?

I've just noticed how this is defined. It reflects the function in the $x$ axis, so when $f$ goes negative $f^-$ goes positive.

Is this right? I've been understanding it as a simple filter: a half-wave rectifier which does not invert, so it is the exact complement of the positive part.

Otherwise it would be called the "absolute value of the negative part" function.

If it is genuinely the way this function is actually defined, please explain this in a "historical note" sort of thing, explaining who it was originated that convention. --prime mover (talk) 22:38, 17 June 2022 (UTC)

I don't know the origin but can partially explain it. This decomposition was introduced to define the integral.

One defines the integral for all $f\ge 0$.

Then, one defines the integral for more general $f$ as:

$\int f := \int f^+ - \int f^-$

where $f = f^+ - f^-$

I guess the idea goes back to Henri Lebesgue.--Usagiop (talk) 23:40, 17 June 2022 (UTC)

What appallingly ill-designed rubbish. So why don't they define it as $f = f^+ + f^-$? Then the negative part will be the negative part and not the absolute value of the negative part. All mathematicians are stupid. --prime mover (talk) 05:05, 18 June 2022 (UTC)

I have invoked the Expand template so we as a mathematical community own to this festering pile of liquid faecal matter.

I think the design is natural. It is similar to the fact that the imaginary part of $a+ib$ is just $b$, and not $ib$. The latter is also less useful. --Usagiop (talk) 09:04, 18 June 2022 (UTC)

Whoa, calm down here. We are (ultimately) just trying to reduce the integral of an arbitrary function to the (simpler) integral of positive functions. And yes, for that definition to work, the negative part of a function needs to be... positive. Or would you like to create the "ill-designed rubbish" of repeating literally every theorem about positive functions for a function which is everywhere nonpositive? Didn't think so either.

But I agree that a statement to that effect could help to elucidate the definition, since it defies immediate intuition. — Lord_Farin (talk) 07:26, 18 June 2022 (UTC)

One immediate problem I see is that if one were to explore properties of mappings in a different context to measure theory, they might want to consider the "negative part" where such a concept does require that the negative part of $f$ be defined using $\map \min {0, \map f x}$.

Hence it might be worth renaming this Definition:Negative Part (Measure Theory). --prime mover (talk) 07:36, 18 June 2022 (UTC)

In fact I have just cracked open 1970: Arne Broman: Introduction to Partial Differential Equations and he does define the "negative part" as $\forall x \in X: \map {f^-} x := \begin {cases} \map f x & : \map f x \le 0 \\ 0 & : \map f x > 0 \end {cases}$, going on to develop the theory in parallel with Lebesgue's work by inserting judicious negative signs where appropriate.

Hence yes, I do think it is not necessarily absurd to present the theory using this more intuitively natural definition of Definition:Negative Part. --prime mover (talk) 07:41, 18 June 2022 (UTC)

Renaming to Definition:Negative Part (Measure Theory) would be better. I don't think this belongs to Real Analysis, though not 100% sure.--Usagiop (talk) 09:18, 18 June 2022 (UTC)

How is that for a more-or-less acceptable compromise? --prime mover (talk) 08:56, 18 June 2022 (UTC)

Looking good, thanks! — Lord_Farin (talk) 09:18, 18 June 2022 (UTC)