# What is the difference between an accuracy measure and an error metric?

The two concepts are distinct in measure theory. Nonetheless, moving out from measure theory, the two terms are often used interchangeably. To most forecasters, especially forecast practitioners, they both refer to functions of forecast errors, e.g., MBE, MAE, or RMSE.

I am hoping somebody could elaborate on the potential harms and pitfalls when the two terms are mixed up.

A measure $$\mu$$ on a set $$X$$ is a mapping $$\mu:\mathcal{A}\rightarrow[0,\infty]$$ defined on a $$\sigma$$-algebra $$\mathcal{A}$$ that satisfies non-negativity, null empty set, and $$\sigma$$-additivity, that is $$\mu(A) \ge 0 \,\, \forall \,\, A \in \mathcal{A}$$, $$\mu( \emptyset)=0$$, and $$\mu( \sqcup_{j \in \mathbb{N}} A_j) = \sum_{j \in \mathbb{N}} \mu(A_j)$$, where symbol $$\sqcup$$ denotes disjoint union. On the other hand, a metric is a distance measure $$d:X\times X \rightarrow [0,\infty]$$ that satisfies definiteness, symmetry, and triangle inequality, that is $$d(x,y) = 0$$ iff $$x = y$$, $$d(x,y) = d(y,x)$$, and $$d(x,y)\le d(x,z) + d(z,y)$$, $$\forall$$ $$x,y,z\in X$$.

I am not so sure when to use the word "measure," and when to use "metric."