Could you help me understand the notation in my professor's lectures

Question

My lecturer in functional analysis constantly write the notation such as $\lambda(dy)$ in integral for a measure $\lambda$ .For example for a operator $$ Tf:=\int exp(x-y)^2f(y)\lambda(dy)$$

I am a little confused with this notation as I only came across the notation like $$ \int exp(x-y)^2f(y)d\lambda(y) $$ before in measure theory.I guess two notations are the same .

Could anyone explains the first notation?

Thanks.

Answer 1

Yes, the notations $\lambda(dy)$ and $d\lambda(y)$ are equivalent. They both mean to integrate the expression as a function of $y$ with respect to the measure $\lambda$.