Suppose we have a utility function $u:\mathbb{R}_+\rightarrow\mathbb{R}$ defined by $u(x)=x$, where $x$ stands for quantity of apples. Suppose we measure the quantity of apples in kg.
How can we ensure that the equation $u(x)=x$ has dimensional homogeneity? It seems that on the left-hand side of this equation we have apples as our dimension and on the right-hand side we have utility... What am I missing?
Not a problem:
$$u(x)=x \text{ kg. apples}\times 1\frac{\text{ util}}{\text{ kg. apple}}$$