On book An introduction to the classification of amenable C*-algebras,p148, it reads
C*-algebras with tracial topological rank $k$ are C*-algebras that can be approximated by C*-subalgebras in $\mathcal I^{(k)}$ in trace (or in "measure").
The class $\mathcal I^{(k)}$ here refers to the class formed with C*-algebras who are unital hereditary subalgebras of C*-algebras of the form $C(X)\otimes F$ where $X$ is a $k$-dimensional finite CW complex and $F$ is a finite dimensional C*-algebra.
And the definition of tracial topological rank, is,
A unital simple C*-algebra $A$ is said to have $TR(A)\leq k$ if for any $\epsilon>0$ and finite subset $\mathcal F$ and nonzero element $a\in A_+$ there exist a nonzero projection $p\in A$ and a subalgebra $A\supseteq B\in \mathcal I^{(k)}$ with $1_B=p$ such that $$\begin{align}&(1)\|px-xp\|<\epsilon\text{ for all } x\in \mathcal F\\ &(2) \text{distance}(pxp,B)<\epsilon \text{ for all }x\in \mathcal F\\&(3) 1-p\text{ is equivalent to a projection in } \mathbf{closure}[aAa]\end{align}$$
I don't get what it means by 'approximated in trace' at all. Is it something like, for every $x\in A$ and tracial state $f$ there is a subalgebra $B\in \mathcal I^{(k)}$ and $y\in B$ such that $f(x-y)$ is sufficiently small?