We present mechanisms to specify and efficiently check, at runtime, assertions that express structural properties and aggregate measures of dynamically manipulated linkedlist data structures. Checking assertions involving the structure, disjointness, and aggregation measures on lists and list segments typically requires linear or quadratic time in the size of the heap. Our main contribution is an incrementalization instrumentation that tracks properties of data structures dynamically as the program executes and leads to orders of magnitude speedup in assertion checking in many scenarios. Our incrementalization incurs a constant overhead on updates to list structures but enables checking assertions in constant time, independent of the size of the heap. We define a general class of functions on lists, called linear measures, which are amenable to our incrementalization technique. We demonstrate the effectiveness of our technique by showing orders of magnitude speedup in two scenarios: one scenario stemming from assertions at the level of APIs of list-manipulating libraries and the other scenario stemming from providing dynamic detection of security attacks caused by malicious rootkits.