Reconciling measure theory change of variables with u-substitution

Symbolically we have $\phi'(t) \, dt = d\phi$ which gives $$ \int_a^b (h \circ \phi)(t) \, \phi'(t) dt = \int_{[a,b]} (h \circ \phi)(t) \, d\phi(t) = \int_{\phi([a,b])} h(x) \, d\phi(\phi^{-1}(x)) = \int_{\phi(a)}^{\phi(b)} h(x) \, dx $$

The measure $\mu$ in your post is defined by $\mu(E) = \int_E \phi'(t) \, dt = \int_E \phi'(t) \, d\lambda(t)$ so $$\frac{d\mu}{d\lambda}(t) = \phi'(t).$$