Why is the notation $\mu(dt)$ in integrals so widely used in Measure Theory?

Question

It seems to me rather ambiguous as it gives the impression of invariance by translation.

Could it be an archaic notation that comes from the Lebesgue measure (for which of course, this notation makes sense)?

It seems to me that $d\mu(t)$ is more precautious with general measures, because one can directly see that it depends on the point $t$.

Answer 1

The notation $\mu(dt)$ helps us to express integrals with respect to a measure in a way that makes the "measuring" aspect of the integral clear.

More explicitly, it has the advantage of making it clear that we are integrating with respect to a measure, rather than integrating over a variable $t$. It also allows us to express the integral in a way that is independent of the measure space (ie. the explicit sample space and sigma algebra), as long as we know how the measure $\mu$ is defined.

The notation was first used by Lebesgue himself in $1901$ in his paper "Sur une généralisation de l'intégrale définie" . Ending the integral with $\mu(dx)$ informs us that the measure $\mu$ is being applied to $dx$ in order to obtain a value that can be integrated over the domain of a function $f$.

The reason that this notation has become standard in measure theory is partly due to the intuition explained above, but also partly because Lebesgue is the single most influential mathematician in Measure Theory (and this is reflected in the fact that we refer to this as the "Lebesgue Integral") and in these cases we typically keep the same notation as is standard unless there is a compelling reason to do otherwise.

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As mentioned in the comments, there are also some valid concerns that many have with the alternative notation $d \mu$ or $d \mu (t)$. One such example comes from the J.P. Serre's lecture on Equidistribution.

Using $d \mu (t)$ is still perfectly acceptable notation that is commonly used, so I don't want to dwell on this too much. But this may help to provide some additional insight into why we have kept around the alternative notation around.

Answer 2

Another perspective comes from probability, where in contexts like conditional probability, we might be interested in parameterised families of measures.

So for example, a transition kernel is a function $\nu(x, B)$, where $B$ is a measurable set, where $x \mapsto \nu(x, B)$ is a measurable function for each $B$ and $B \mapsto \nu(x, B)$ is a probability measure for each $x$.

Then, to integrate against the measure $\nu(x, \;\cdot\;)$, we can write $$ \int f(t)\, \nu(x, \mathrm{d}t), $$ which makes is clear which argument of $\nu$ is the measure, whereas something like $$ \int f(t)\, \mathrm{d}\nu(x, t) $$ is ambiguous.

In a similar vein, you might see in probability things like $$ \mathbf{E}(X | Y = y) = \int x \Pr(X \in \mathrm{d}x | Y = y), $$ which is much harder to see how to write without the $\mu(\mathrm{d} t)$ convention.