TITLE: Difference between a measure and a norm in a euclidean space
QUESTION [7 upvotes]: 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?

REPLY [4 votes]: 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 $λ^2(L)=0$)