  We consider Bernoulli measures $\mu_p$ on the interval $[0,1]$. For the
standard Lebesgue measure the digits $0$ and $1$ in the binary representation
of real numbers appear with an equal probability $1/2$. For the Bernoulli
measures, the digits $0$ and $1$ appear with probabilities $p$ and $1-p$,
respectively. We provide explicit expressions for various $\mu_p$-integrals. In
particular, integrals of polynomials are expressed in terms of the determinants
of special Hessenberg matrices, which, in turn, are constructed from the Pascal
matrices of binomial coefficients. This allows us to find closed-form
expressions for the Fourier coefficients of $\mu_p$ in the Legendre polynomial
basis. At the same time, the trigonometric Fourier coefficients are values of
some special entire function, which admits an explicit infinite product
expansion and satisfies interesting properties, including connections with the
Stirling numbers and the polylogarithm.
