lsusr comments on Examples of Measures

[lsusr]

Here are a few more measures.

Integral measure

Let $f,g$ be two continuous functions defined on the interval $[0,1]$. The integral $I$ of the absolute value of their difference is a measure.

$$I={\int }_0^1|f-g|dx$$

Equality measure

Let $x,y$ be any two mathematical objects. The equality measure equals 0 if $x=y$. Otherwise the equality measure equals 1. (This is equivalent to the side lengths of an $\infty$-dimensional tetrahedron.)

Cartesian measure

Let $m_1,m_2$ be two measures defined on sets $S_1,S_2$, respectively.

• The measure $m_1+m_2$ defined on set $S_1\oplus S_2$ is a measure.

• If $S_1=S_2$ then $m_1+m_2$ is a measure on $S_1,S_2$.

Applying the Cartesian measure to Lebesgue measure gives us taxicab distance. Let $x=(x_1,x_2),y=(y_1,y_2)\in R^2$. Taxicab distance $|x_1-y_1|+|x_2-y_2|$ is a measure.

[DanielFilan]

I think you might be confusing measures with metrics.

[Pattern]

What’s the difference?

[Maxwell Peterson]

Intuitively, a metric outputs how different two things are, while a measure outputs how big something is.

In terms of inputs and outputs: a metric takes two points as input, and outputs a positive real number. A measure takes one set as input, and outputs a positive real number.

[DanielFilan]

A metric gives the ‘distance’ between two points, while a measure gives the ‘weight’ of a subset (quotes are scare quotes).

[lsusr]

Thanks…I’m just going to quietly retract the original comment.