The authors studied various relationships of the Hausdorff dimension, entropy dimension and \(L^p\)-dimension of a measure. The main results of the paper are: Let \(0<p<1<q\), then \[ {\underline{\dim}}_q\mu\leq\dim_* \mu \leq {\underline{\dim}}_e\mu\leq {\overline{\dim}}_e\mu\leq\dim^*\mu\leq {\overline{\dim}}_p, \] where \({\underline{\dim}}_q\), \(\dim_*\mu\), \({\underline{\dim}}_e\mu\), \({\overline{\dim}}_e\mu\), \(\dim^*\mu\), \({\overline{\dim}}_p\) are, respectively, the lower \(L^q\)-dimension, lower Hausdorff dimension, lower entropy dimension, upper entropy dimension, upper packing dimension and upper \(L^p\)-dimension of the measure \(\mu\). The result extends a well-known result of Young: if the local dimension of \(\mu\) exists and is equal to \(\alpha\), \(\mu\)-a.e., then \(\dim_e\mu=\dim_H\mu=\alpha\), without assuming that the local dimension of \(\mu\) exists \(\mu\)-a.e.