Let K be a trace class
operator on L2(X,ℳ,μ) with integral kernel K(x,y) ∈ L2(X × X,μ × μ). An
averaging process is used to define K on the diagonal in X × X so that the
trace of K is equal to the integral of K(x,x), generalizing results known
previously for continuous kernels. This formula is also shown to hold for
positive-definite Hilbert-Schmidt operators, thus giving necessary and sufficient
conditions for the traceability of positive integral kernels. These results make
use of Doob’s maximal theorem for martingales and generalize previous
results obtained by the author using Hardy-Littlewood maximal theory when
X ⊂Rn.
