Regardless of the technical merits of the Lebesgue integral, measure is a
fundamental analytic notion, as fundamental in analysis as topology is in
topology.

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One good way to introduce measure is to introduce integration with respect to
a measure (i.e., start working with it before defining it).

A good example is torque of a rod, as this naturally has continuous and singular
measures, namely:
a rod with some mass (distributed uniformly at first, then non-uniformly)
has torque given by integration with respect to a continous measure;
a massless rod with some masses hung on it has torque given by integration
with respect to a singular measure (supported at the points where the masses
are hung);
a rod with mass, and with some masses hung on it, has a mass distribution with
a continuous and singular part.
This leads in to the Lebesgue decomposition Theorem, but that's a bit beyond
the introductory level.

This is a way of smoothing over the continuous/discrete distinction that
throws people for a loop (e.g., in discrete probability and working with
series, you're integrating over a discrete space), as people are used to
working with this.

(measures are a "better" generalization of continous function than
"discontinuous function" for the purposes of analysis, as you can still
integrate them and they behave kinda nicely)

exercises:
prove that the measure of $(c,d)$ (call it $\mu = \mu(c,d)$) is $d-c$,
by noting:
$(c,d) \subset [c,d]$ which has measure $d-c$,
so we should have $\mu \leq d-c$ (note that despite being a proper
containment, it's not a strict inequality), and on the other side we have
$[c+\epsilon,d-\epsilon] \subset (c,d)$, so $d-c-2\epsilon \leq \mu$,
for all $\epsilon$, so
$d-c \leq \mu \leq d-c$, so $\mu = d-c$.

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Again:
measureable set is as fundamental a notion as open set

They're seen as somewhat weird by some at first b/c they're less
familiar and harder to describe.
The originally showed up artificially (in Cantor's descriptive set
theory),
but show up very naturally in dynamics.

Indeed, the analogy between measure and topology is quite rich.
