The measure
μ
φ
\mu _\varphi
induced by a bounded analytic function
φ
\varphi
on the unit disk
U
U
may be defined by
μ
φ
(
E
)
=
m
(
φ
−
1
(
E
)
)
\mu _\varphi (E)=m(\varphi ^{-1}(E))
, where
m
m
is normalized Lebesgue measure on
∂
U
\partial U
. We discuss the problem of characterizing such measures, and produce some necessary conditions which we conjecture are sufficient. Our main results are a construction showing that our conjectured sufficient conditions are sufficient for a measure to be weakly approximable by induced measures, and a construction of a function
φ
\varphi
, not a constant multiple of an inner function, whose induced measure is rotationally symmetric. This function is not inner, but satisfies
∫
φ
(
e
i
θ
)
m
φ
(
e
i
θ
)
n
¯
d
θ
2
π
=
0
\int \varphi \left (e^{i\theta }\right )^m\overline {\varphi \left (e^{i\theta }\right )^n} \frac {d\theta }{2\pi }=0
if
m
≠
n
m\ne n
, thus answering a question posed by Walter Rudin.