4. Given $f(x)=a \sin x+b \sqrt[3]{x}+c \ln \left(x+\sqrt{x^{2}+1}\right)+1003(a, b, c$ are real numbers $)$, and $f\left(\lg ^{2} 10\right)=1$, then $f(\lg \lg 3)=$ $\qquad$ .
4. 2005

Notice that $\ln \left(x+\sqrt{x^{2}+1}\right)$ is an odd function, so $f(x)-1003$ is an odd function. Also, $\lg \log _{3} 10=\lg \frac{\lg 10}{\lg 3}=-\lg \lg 3$, thus it is easy to find that the result should be 2005.