Abstract: This paper proved that for a quantized metric space (A, L) coming from an action a of a compact group G on a C*-algebra A, there exists a sequence of finite-dimensional quantized metric spaces (A_n, L⁽ⁿ⁾) such that (A, L) is the limit of (A_n, L⁽ⁿ⁾) with respect to quantized Gromov-Hausdorff distance.

Key words: quantum Gromov-Hausdorff distance, ergodic action, length function, quantized metric space, quantum Gromov-Hausdorff distance, ergodic action, length function