Consider the model $$\partial _{t}\mu + \partial_{x}(b(t,\mu)\mu)+c(t,\mu) \mu=0,~~~\mu \in \mathcal{M}^{+}(\mathbb{R}^{+}), t \in [0,T], x \in \mathbb{R}^{+} $$ $$ \mu(0)=\mu_{0} $$ where $ \mu (t)$ is the measure determining the distribution of the population with respect to the structural variable $x$ and $b,c$ are vital rates.

I want to know what it means for a solution to be measure-valued i.e. solution is a measure? If the solution (in weak sense) is measurable function then it makes sense to me but solution as a measure really confusing me, because measure is defined on a set and solution we usually need at every point of domain.

Consider the model $$\partial _{t}\mu + \partial_{x}(b(t,\mu)\mu)+c(t,\mu) \mu=0,~~~\mu \in \mathcal{M}^{+}(\mathbb{R}^{+}), t \in [0,T], x \in \mathbb{R}^{+} $$ $$ \mu(0)=\mu_{0} $$ where $ \mu (t)$ is the measure determining the distribution of the population with respect to the structural variable $x$ and $b,c$ are vital rates.

I want to know what it means for a solution to be measure-valued i.e. solution is a measure? If the solution (in weak sense) is measurable function then it makes sense to me but solution as a measure is really confusing me, because measure is defined on a set and solution we usually need at every point of domain.