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I'm currently starting to study measure theory and the definitions I've seem up to now are:

A measurable space is a pair $(X,M)$ being $X$ a non empty set and $M$ a $\sigma$-algebra of subsets of $X$ which are called measurable sets.

A measure on the measurable space $(X,M)$ is a function $\mu : M\to [0,+\infty]$ satisfying:

  • $\mu(\emptyset)=0$,
  • If $\{E_n \in M : n\in \mathbb{N}\}$ is a countable collection of pairwise disjoint measurable sets then $$\mu\left(\bigcup_{n\in \mathbb{N}}E_n\right)=\sum_{n=1}^\infty \mu(E_n).$$

This indeed make sense since the idea here is that $\mu$ is a generalized notion of the "size" of the measurable sets.

The point is that I'm aware that there are measures which are not real valued like the one defined above. For example, I've already heard about complex measures. Another quite interesting example is the projector-valued measures which appear on the spectral theorem in Functional Analysis.

Those projector-valued measures don't even take values on any set of numbers. They take value on a subset of the algebra of operators on a Hilbert space.

Now, at first I find it quite intuitive the definition of measure given above, but I can't see how these other measures arise and how they are meaningful.

So, how those measures which are not real-valued arise? How they relate to this definition? And I think more importantly, what is the intuition behind then? The measure defined above is meant to generalize the idea of size of a set, but those other measures what they intend to be? They can't be about sizes, since they can take values on sets which are not of numbers. So what is the point with them?

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Usually, the codomain of a measure can be $[0,+\infty]$, $\mathbb{R} \cup \{-\infty \}$, $\mathbb{R} \cup \{+\infty \}$, $\mathbb{C}$ and any Banach space. The rest of the definition of measure remains the same. The usual names are:

  • measure, if the codomain is $[0,+\infty]$;
  • signed measure, if if the codomain is $\mathbb{R} \cup \{-\infty \}$ or $\mathbb{R} \cup \{+\infty \}$;
  • complex measure, if the codomain is $\mathbb{C}$;
  • vector valued measure or vector measure, if the codomain is a Banach space

Since the continous linear transformation in a Hilbert Space form a Banach space, projector-valued measures which appear on the spectral theorem in Functional Analysis is a special case of vector valued measure. This special case is sometimes called spectral measures.

The intuition behind those concepts is always the same: "to measure". Let us use a litle Physics here. To measure the total electric charge of a region of the space you will probably want a signed measure. To measure the total magnetic field of a region of the space, you will probably want a vector valued measure (one with codomain $\mathbb{R}^3$). Of course, those as just informal motivation.

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