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Monitoring transit traffic at one or more points in a network is of interest to network operators for reasons of traffic accounting, debugging or troubleshooting, forensics, and traffic engineering. Previous research in the area has focused on deriving a placement of monitors across the network towards the end of maximizing the monitoring utility of the network operator for a given traffic routing. However, both traffic characteristics and measurement objectives can dynamically change over time, rendering a previously optimal placement of monitors suboptimal. It is not feasible to dynamically redeploy/reconfigure measurement infrastructure to cater to such evolving measurement requirements. We address this problem by strategically routing traffic sub-populations over fixed monitors. We refer to this approach as MeasuRouting. The main challenge for MeasuRouting is to work within the constraints of existing intra-domain traffic engineering operations that are geared for efficiently utilizing bandwidth resources, or meeting Quality of Service (QoS) constraints, or both. A fundamental feature of intra-domain routing, that makes MeasuRouting feasible, is that intra-domain routing is often specified for aggregate flows. MeasuRouting, can therefore, differentially route components of an aggregate flow while ensuring that the aggregate placement is compliant to original traffic engineering objectives. In this paper we present a theoretical framework for MeasuRouting. Furthermore, as proofs-of-concept, we present synthetic and practical monitoring applications to showcase the utility enhancement achieved with MeasuRouting.