  It is well-known the Lebesgue \cite{Lebesgue, Zygmund} test for trigonometric
Fourier series. Taberski \cite{Taberski1, Taberski2} considered real-valued
Lebesgue locally integrable functions $f$, such that \begin{equation*} \lim_{T
\to \infty} \frac{1}{T} \int_{T}^{T+c} |f(t)| \, dt\ =0; \quad \lim_{T \to
\infty} \frac{1}{T} \int_{-T-c}^{-T} |f(t)| \, dt \ =0 \end{equation*} for
every fixed $c>0$. For this class of functions, he defined generalized
Dirichlet's integrals. Besides, Taberski \cite{Taberski1,Taberski2}
investigated problems of convergence and $(C,1)$-summability of these
integrals. In this paper, the analogous of the Lebesgue test for the
generalized Dirichlet's integrals is proved.
