Is it applicable to find an expression for magnitude of the Dirac Delta Function?

Question

Given $x(t)=\delta\left(t-\dfrac{\pi}{2}\right)$
Is it correct to say that $|x(t)|=\delta\left(t-\dfrac{\pi}{2}\right)$ ?
Or can't we just apply magnitude to Delta Function? Thanks in advance.

Answer 1

As opposed to what phisicist think, the Dirac delta is not a function. The Dirac delta is a measure. That is it is defined on a $\sigma$-algebra.

In simple terms $\delta_0(I)=1$ if and only if $0\in I$ and $\delta_0(I)=0$ otherwise, for some class of subsets $I$ of $\mathbb{R}$.

So your question becomes: can we give meaning to the absolute value/magnitude of a measure?

The answer is yes and is the same idea that is behind the absolute value of a function: if $f^+$ and $f^-$ are two positive functions and $$f=f^+-f^-$$ then $$|f|=f^++f^-.$$ Any function can be written as the difference of a positive and negative part.

Less trivially any finite measure can be written as the difference of two finite positive measures: $\mu=\mu^+-\mu^-$. Then what you speak of is the total variation measure of a measure (a little bit of abuse): $$|\mu|=\mu^++\mu^-.$$ As such the total variation measure of a positive measure such as the Dirac delta is just the Dirac delta itself: $|\delta_0|=\delta_0$.