In the era of big data, data stream is a common data model with characteristics such as orderly, massive and time-varying. Fractal is an important feature of many complex systems, and is mainly represented by fractal dimension. Data stream can be viewed as a dynamic and complex system, and its fractal dimension should also have characteristics of dynamic, time-varying and multi-granularity. This paper presents a method of measuring multi-granularity and time-varying fractal dimension on a data stream based on discrete wavelet transform. The method can simultaneously measure the time-varying fractal dimension on a data stream by using the summary information from wavelet transforming of the data stream saved in a multi-granularity wavelet transforming tree in memory. This method has low computational complexity, and effectively reveals the evolution of a data stream. Experimental results show that it can effectively monitor the time-varying characteristic of fractal dimension on a data stream at different granularity.