SINGAPORE/MILAN, June 5 (Reuters) - World shares rose on Wednesday and the dollar steadied with a European Central Bank policy meeting coming into focus following soft U.S. labour market data that firmed up bets of a September rate cut by the Federal Reserve.

Worries about a cooling U.S. economy, however, kept a lid on risk appetite. The focus in Asia stayed on Indian markets, with stocks rising after Tuesday's plunge as voting results showed a slimmer-than-expected victory margin for PM Narendra Modi.

The ECB meets on Thursday and money markets price in an almost certain chance of a first interest rate cut, but there is uncertainty about the future path for rates in the euro zone.

"I have a positive view on tomorrow's cut because it marks the end of an era of rate hikes that began two years ago," said Carlo Franchini, head institutional clients at Banca Ifigest.

"Now, we'll need to see the impact that rate cuts will have on domestic demand and the economic recovery."

Data on Wednesday showed euro zone business activity expanded at its quickest rate in a year in May as growth in services industry outpaced contraction in manufacturing.

Across the Atlantic, eyes were on the Bank of Canada's policy meeting later on Wednesday with traders expecting the central bank to begin its rate-cutting cycle.

The MSCI world equity index < MIWD00000PUS>, which tracks shares in 49 countries, added 0.05% by 0802 GMT, supported by a positive open in Europe and gains in Asia.

The pan-European STOXX 600 index was up 0.5% and the MSCI's broadest index of Asia-Pacific shares outside Japan rose 0.9%. The Nikkei in Tokyo fell 0.9% as the renewed strength in Japanese yen weighed.

Data on Tuesday showed U.S. job openings fell more than expected in April to the lowest level in more than three years, a sign that labour market conditions are softening.

The data emboldened bets of Fed rate cuts this year, with markets pricing in 45 basis points of easing. Traders are pricing in a 65% chance of a rate cut in September, compared with 46% a week earlier, CME FedWatch tool showed.

"I think there's a strange trade off between those who like that data as encouraging thoughts of rate cuts and those who are looking at things going on and that are not great news," said Rob Carnell, ING's regional head of research for Asia Pacific.

"Asia wants to find a little bit of good news in the very mixed run out of numbers that we're seeing."

The market focus will also be on the U.S payrolls report due on Friday. Wall Street futures pointed to gains of 0.1-0.2% for the S&P 500 and X% for the tech-heavy Nasdaq.

Benchmark 10-year note yields were at 4.3552% on Wednesday, after hitting an almost three-week low of 4.314 on Tuesday following the jobs data.

Germany's 10-year government bond yield, the benchmark for the euro zone, was steady at 2.543% after its sharpest two-day drop since March.

The dollar index, which measures the U.S. currency against six peers, was 0.1% higher at 104.29, just above the near two-month low of 103.99 it hit on Tuesday.

The dollar's relentless strength in the recent past will make way for minor weakness over the next 12 months, according to a Reuters poll of strategists who generally agreed the dollar was overvalued.

The dollar's retreat helped the yen strengthen to a more than two-week high of 154.55 per dollar on Tuesday. On Wednesday, it weakened to 156.11.

India's Nifty 50 rose 2.4% in volatile trading after sliding nearly 6% on Tuesday, its worst session in four years, with foreign investors selling roughly $1.5 billion of shares.

Modi's ruling Bharatiya Janata Party lost an outright majority in parliament for the first time in a decade and is dependent on its regional allies to get past the half-way mark required to run the world's largest democracy.

In commodities, oil prices hovered near four-month lows as traders weighed an OPEC+ decision to boost supply later this year and an increase in U.S. crude and fuel stocks.

Brent crude futures were last at 77.71 per barrel, up 0.25%, while U.S. West Texas Intermediate crude futures traded at $73.41 a barrel, up 0.2%.

(Reporting by Ankur Banerjee in Singapore and Danilo Masoni in Milan Editing by Christina Fincher)