Difference between a measure and a norm in a euclidean space

Question

For example: If we have a line in $\mathbb{R^2}$, would the length of the line be its norm or measure?

Could someone please explain the difference?

EDIT: Is there any scenario, where "norm" and "measure" are equivalent terms?

Answer 1

In a sense the length of line is a nonlinear variant of a norm. Let me explain what I mean. Imagine you have a straight line segment, represented by some vector $a\in\mathbb{R}^2$, i.e. $L=\{v\in\mathbb{R}^2:\ v=\lambda a,\ 0\leq\lambda\leq 1\}.$ Then the length of $L$ coincides with the norm of $a$. If you now have a curved line-segment, you define the length of it by building upon this 'linear' notion of length. One way to do it, would be the Hausdorff measure (see e.g. https://en.wikipedia.org/wiki/Hausdorff_measure). For this defintion the notion of diameter is paramount, which can be interpreted as the longest straight line segment fitting into a set.

You also asked for a scenario in which norm and measure would be equivalent. The first thing which came to my mind would be the definition of perimeter by Caccioppoli. (see e.g. https://en.wikipedia.org/wiki/Caccioppoli_set) This perimeter measures the length of the boundary of a set $E$. It can be defined as the total variation of the indicator function of the set $E$ (see e.g. https://en.wikipedia.org/wiki/Bounded_variation#BV_functions_of_several_variables), which is a seminorm on the respective function space.

Answer 2

Norm is mainly used in the context of vector spaces, i.e. (vaguely speaking, vector space is) a set equipped with a structure that enables us to sum up elements of the set and multiply them with scalars (e.g. real numbers). Measure is mostly used in the context of sets which are not required to have any additional structure. Yet, to define the notion of measure we need to require certain rules (about the open subsets of the given set and their unions and intersections) to hold.

Basic example of a norm would be the Euclidean norm (square all the entries of a given vector, sum them up and take the square root - classic). Basic example of a measure would be the length of an interval, area of a two dimensional region, or a volume of some higher-dimensional object. But the objects we want to measure are not just these elementary shapes, but also their unions and intersections (so thing may get quite complicated as far as general element to be measured is concerned - unlike with the case of vector spaces where the element may be uniquely represented with a sequence of numbers (components of the vector)).

Let our set be denoted by $S$. Both the norm and the measure are a certain functions on a set under consideration. A bit more precisely, norm (denoted $|| \cdot ||$) takes only one argument from $S$ and spits out a non-negative number., i.e. it is a mapping $|| \cdot || \colon S \rightarrow \mathbb{R}$ satisfiying certain rules such as the norm of a zero vector to be zero. On the other hand a measure takes a subset of $S$ and spits out a non-negative number. So $\mu \colon \Omega(S) \rightarrow \mathbb{R}$ is a function considered on a certain set $\Omega(S)$ consisting of subsets of $S$ and we require some axioms about $\Omega(S)$ to hold (so actually some structure is hidden somewhere else anyway yet different structure then the vector spaces have) in order for the measure to be definable nicely. Moreover, other axioms must be satisfied by $\mu$ to be a function called measure (e.g. that empty set has zero measure).

Note: The axioms of a measure does not require for non-empty sets to be of zero measure - unlike the case of a norm where non-zero vector has always non-zero norm... well, actually we can consider a type of norm which assigns zero to a non-zero vector but those are usually called pseudo-norms.

Answer 3

I think that if we see the definitions of norm and mesure we can make the situation clearer.

A norm is usualy defined in a vector space and is a function which takes a vector and spits out some non negative number. It is also required to have some certain properties that make it behave like a length function. This axioms are

• $||x||>0 $ if $x \ne 0 $ and $||x||=0$ if $x=0$ (as elements of the vector space you have)
• $||x+y||\leq||x||+||y||$ for all vectors (this is the analogue of the triangle inequallity)
• $||αx||=|α|\cdot||x||$ for all vectors $x$ and scallars $α$

The most usual example is the Euclidian norm in $\mathbb{R^2}$ which given a vector $u=(x,y)$ gives $||u||=\sqrt{x^2+y^2}$

Now a measure is a function which takes sets (which when you are in a familiar setting like $\mathbb{R^2}$ can be thought as shapes) and spits out non-negative numbers.The corresponding axioms are:

• $μ(Α)\geq0$ for all sets A
• the empty set has measure zero
• If we have a family of disjoint sets $A_i$ then $μ(\cup_i^{\infty} A_i)=\sum_i^{\infty}μ(A_i)$

The most usual meausre is the Lebesgue Measure which , in good sets coincides with our sense of length in $\mathbb{R^1}$, area in $\mathbb{R^2}$ and volume in $\mathbb{R^3}$

Now for your question:The answer depends on how you view your objects. If you want to see the line as a vector (with , say, the start to the origin) then the norm would be the length of the line.

Now it gets trickier because the standar measure in $\mathbb{R^2}$ is area so it is expected (and thats what happens does indeed) for every line, segment or infinite to have measure (area) 0. But if you identify the line with $\mathbb{R^1}$ then you would have as measure what we usualy understand as length.

To make this clear. Say you have the line going from $(0,0)$ to $(1,1)$. This can be represented as

• the vector: $u=(1,1)$
• the subset of $\mathbb{R^2}$: $L=\{$all points $p=(x,y)$ of $\mathbb{R^2}$ which are of the form $(x,x)$ for $0\leq x\leq1\}$

Then the norm $||u||=\sqrt2$ and by identifying $L$ with the set $L'=[0,\sqrt2]$ you have $λ^1(L')=\sqrt2$ with $λ'$ being the measure of $\mathbb{R}$ (note that as we said above the Lebesgue measure of $\mathbb{R^2}$ would give the area of the line so $λ^2(L)=0$)